Inclusion Criteria:  Ability to comply with the study protocol, in the investigator's judgment Histologically or cytologically confirmed solid tumors (cutaneous melanoma, non-small cell lung cancer, renal cell carcinoma, endometrial cancer, ovarian cancer, MSI-H solid tumors (deficient mismatch repair system or other solid tumors) that have progressed on or refractory to standard of care therapies for their disease  Phase I dose escalation phase  Advanced metastatic or recurrent solid tumors (where pembrolizumab is approved and/or have shown efficacy) that have progressed on or refractory to standard of care therapies for their disease Prior anti-PD1 or PDL1 therapy is allowed Prior anti-CTLA4 therapy is allowed if had anti-CTLA4 free interval of 6 months or more At least one prior line of therapy in the dose escalation phase  Phase Ib dose expansion cohorts  Platinum-resistant recurrent ovarian cancer (recurred within 6 months or less of prior platinum therapy) or post-PD1/PDL1 MSI-H cancers (mismatch repair deficient tumors). Patients in the ovarian cancer cohort must have received chemotherapy plus bevacizumab unless bevacizumab is contra-indicated or considered risky per treating physician. Prior 1-5 lines of therapy for dose expansion Prior anti-PD1/PDL1 therapy is allowed in the MSI-H cohort but not in the ovarian cohort Prior anti-CTLA4 therapy is not allowed. Primary platinum- refractory cancers are excluded in the dose expansion (progressed during or within 3 months of primary platinum therapy)  Measurable disease per RECIST v1.1  o Previously irradiated lesions can be considered as measurable disease only if progressive disease has been unequivocally documented at that site since radiation.  Availability of a representative tumor specimen for exploratory biomarker research ECOG Performance Status of 0-1 Life expectancy ≥ 6 months  Adequate hematologic and end-organ function, defined by the following laboratory test results, obtained within 14 days prior to initiation of study treatment:  ANC ≥ 1.5 × 109/L (1500/L) without granulocyte colony-stimulating factor support Platelet count ≥ 100 × 109/L (100,000/L) without transfusion Hemoglobin ≥ 90 g/L (9 g/dL) o Patients may be transfused to meet this criterion.  AST, ALT, and alkaline phosphatase (ALP) ≤2.5 × upper limit of normal (ULN), with the following exceptions:  Patients with documented liver metastases: AST and ALT ≤ 5 × ULN Patients with documented liver or bone metastases: ALP ≤ 5 × ULN  Serum bilirubin ≤1.5 × ULN with the following exception:  o Patients with known Gilbert disease: serum bilirubin ≤3 × ULN  Serum creatinine ≤1.5 × ULN} Serum albumin ≥ 25 g/L (2.5 g/dL) For patients not receiving therapeutic anticoagulation: INR or aPTT × 1.5 × ULN o For patients receiving therapeutic anticoagulation: stable anticoagulant regimen Patients with a positive HIV disease are eligible provided they are stable on anti-retroviral therapy, have a CD4 count ≥ 200/μL, and have an undetectable viral load No known Hepatitis infection  For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods, and agreement to refrain from donating eggs, as defined below:  Women must remain abstinent or use contraceptive methods with a failure rate of < 1% per year during the treatment period and for 5 months after the final dose of pembrolizumab and E7777. Women must refrain from donating eggs during this same period. A woman is considered to be of childbearing potential if she is postmenarchal, has not reached a postmenopausal state (≥ 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). Examples of contraceptive methods with a failure rate of < 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of contraception.  For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below:  With a female partner of childbearing potential or pregnant female partner, men must remain abstinent or use a condom during the treatment period and for 5 months after the final dose of pembrolizumab + E7777 to avoid exposing the embryo. Men must refrain from donating sperm during this same period. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of preventing drug exposure  Exclusion Criteria:  Patients who meet any of the following criteria will be excluded from study entry:  History of leptomeningeal disease. Patients with brain metastasis are allowed if their brain metastasis was adequately treated with surgery or radiation or both, have been stable for at least 6 months and not on steroids.  Uncontrolled tumor-related pain  Symptomatic lesions (e.g., bone metastases or metastases causing nerve impingement) amenable to palliative radiotherapy should be treated prior to enrollment. Patients should be recovered from the effects of radiation. There is no required minimum recovery period. Asymptomatic metastatic lesions that would likely cause functional deficits or intractable pain with further growth (e.g., epidural metastasis that is not currently associated with spinal cord compression) should be considered for loco-regional therapy if appropriate prior to enrollment.  Uncontrolled pleural effusion, pericardial effusion, or ascites requiring recurrent drainage procedures (once monthly or more frequently)  o Patients with indwelling catheters (e.g., PleurX™) are allowed.  Uncontrolled or symptomatic hypercalcemia (ionized calcium > 1.5 mmol/L, calcium > 12 mg/dL or corrected serum calcium > ULN)  Active or history of autoimmune disease or immune deficiency, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjögren syndrome, Guillain-Barré syndrome, or multiple sclerosis for a more comprehensive list of autoimmune diseases and immune deficiencies), with the following exceptions:  Patients with a history of autoimmune-related hypothyroidism who are on thyroid-replacement hormone are eligible for the study. Patients with controlled Type 1 diabetes mellitus who are on an insulin regimen are eligible for the study.  Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible for the study provided all of following conditions are met:  Rash must cover < 10% of body surface area Disease is well controlled at baseline and requires only low-potency topical corticosteroids No occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high-potency or oral corticosteroids within the previous 12 months  History of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on screening chest computed tomography (CT) scan  o History of radiation pneumonitis in the radiation field (fibrosis) is permitted.  Active tuberculosis Significant cardiovascular disease (such as New York Heart Association Class II or greater cardiac disease, myocardial infarction, or cerebrovascular accident) within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina Major surgical procedure, other than for diagnosis, within 4 weeks prior to initiation of study treatment, or anticipation of need for a major surgical procedure during the study History of concurrent second primary malignancy other than the primary cancer within 3 years prior to screening, with the exception of malignancies with a negligible risk of metastasis or death (e.g., 5-year OS rate > 90%), such as adequately treated carcinoma in situ of the cervix, non-melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or Stage I uterine cancer Severe infection within 4 weeks prior to initiation of study treatment, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia  Treatment with therapeutic oral or IV antibiotics within 2 weeks prior to initiation of study treatment  o Patients receiving prophylactic antibiotics (e.g., to prevent a urinary tract infection or chronic obstructive pulmonary disease exacerbation) are eligible for the study.  Prior allogeneic stem cell or solid organ transplantation Any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications Treatment with a live, attenuated vaccine within 4 weeks prior to initiation of study treatment, or anticipation of need for such a vaccine during pembrolizumab treatment or within 5 months after the final dose of pembrolizumab Has a known history of human immunodeficiency virus (HIV) infection or HIV test (+) in screening test. No HIV testing is required unless mandated by local health authority. Patients with HIV infection with following exception are allowed: HIV-infected patients on effective anti-retroviral therapy with viral load within 6 months of enrollment are eligible for this trial. Has a known history of Hepatitis B (defined as Hepatitis B surface antigen [HBsAg] reactive) or known active Hepatitis C virus (defined as HCV RNA [qualitative] is detected) infection except following situation: Patients with a known history of Hepatitis B (defined as Hepatitis B surface antigen [HBsAg] reactive) or known active Hepatitis C virus (defined as HCV RNA [qualitative] is detected) infection are allowed to be included if: participant on a stable dose of antiviral therapy, HBV viral load below the limit of quantification. HCV viral load below the limit of quantification. Treatment with investigational therapy within 28 days prior to initiation of study treatment  Prior treatment with CD137 agonists or immune checkpoint blockade therapies, including anti-CTLA-4, anti-PD-1, and anti-PD-L1 therapeutic antibodies except what is specified in the inclusion criteria.  o Prior anti-PD1 or anti-CTLA4 therapy is allowed in the dose escalation cohort and prior PD1/PDL1 inhibitors are allowed in the MSI-H expansion cohort.  Treatment with systemic immunostimulatory agents (including, but not limited to, interferon and interleukin 2 [IL-2]) within 4 weeks or 5 half-lives of the drug (whichever is longer) prior to initiation of study treatment  Treatment with systemic immunosuppressive medication (including, but not limited to, corticosteroids, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-TNF-α agents) within 2 weeks prior to initiation of study treatment, or anticipation of need for systemic immunosuppressive medication during study treatment, with the following exceptions:  Patients who received acute, low-dose systemic immunosuppressant medication or a one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible for the study after Principal Investigator confirmation has been obtained. Patients who received mineralocorticoids (e.g., fludrocortisone), corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenal insufficiency are eligible for the study. History of severe allergic anaphylactic reactions to chimeric or humanized antibodies or fusion proteins Known hypersensitivity to Chinese hamster ovary cell products or to any component of the pembrolizumab formulation Known allergy or hypersensitivity to any component of the E7777 formulation Pregnancy or breastfeeding, or intention of becoming pregnant during study treatment or within 6 months for pembrolizumab and E7777 after the final dose of study treatment o Women of childbearing potential must have a negative serum pregnancy test result within 14 days prior to initiation of study treatment.