Inclusion Criteria:  Female participants who are at least 18 years of age. Signed written informed consent. A histology confirming diagnosis of high grade serous ovarian/peritoneal/fallopian tube cancers and platinum-resistant disease as defined as disease progression on a platinum-containing agent or recurrent within 180 days of prior dose of a platinum-containing chemotherapeutic regimen will be enrolled in this study. Pathology must have been reviewed at MD Anderson.  A female participant is eligible to participate if at least one of the following conditions applies:  Not a woman of childbearing potential (WOCBP) OR A WOCBP who agrees to follow the contraceptive guidance during the treatment period and for at least 5 months after the last dose of study treatment. Measurable disease is present as defined by modified Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1 criterion, and there is disease present in the peritoneal cavity or retroperitoneal lymph nodes. Disease outside the peritoneal cavity is allowed as long as metastases are present within the peritoneal cavity or retroperitoneum. Adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as BP less than or equal to 150/90 mmHg at screening and no change in antihypertensive medications within 1 week prior to the start of treatment. Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. Ability to understand and willingness to sign informed consent form prior to initiation of the study and any study procedures. Participants willing to undergo intraperitoneal port placement and scheduled peritoneal fluid and peripheral blood draws. Absolute neutrophil count (ANC) >= 1500/uL (within 10 days prior to start of study treatment) Platelets >= 100,000/uL (within 10 days prior to start of study treatment)  Hemoglobin >= 9.0 g/dL (transfusion is allowed) (within 10 days prior to start of study treatment)  Criteria must be met without erythropoietin dependency and without packed red blood cell (pRBC) transfusion within last 2 weeks of the screening test. Participants may be on a stable dose of erythropoietin (>= approximately 3 months). Creatinine =< 1.5 x upper limit of normal (ULN) OR creatinine clearance (CrCl) >= 30 mL/min for participants with creatinine > 1.5 x ULN (within 10 days prior to start of study treatment) Total bilirubin =< 1.5 x ULN OR direct bilirubin =< ULN for participants with total bilirubin levels > 1.5 x ULN (within 10 days prior to start of study treatment) Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase [SGOT]) and alanine aminotransferase (ALT)(serum glutamic pyruvate transaminase [SGPT]) =< 2.5 x ULN (=< 5 x ULN for participants with liver metastases) (within 10 days prior to start of study treatment) International normalized ratio (INR) OR prothrombin time (PT) =< 1.5 x ULN unless participant is receiving anticoagulant therapy as long as PT or activated partial thromboplastin time (aPTT) is within therapeutic range of intended use of anticoagulants (within 10 days prior to start of study treatment)  Exclusion Criteria:  Is pregnant, breastfeeding, or expecting to conceive within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment. If a WOCBP who has a positive urine pregnancy test within 72 hours prior to randomization that cannot be confirmed as negative, a serum pregnancy test will be required. Has received prior therapy with lenvatinib Has received systemic anti-cancer therapy including investigational agents (e.g. anti-vascular endothelial growth factor (VEGF) or immune checkpoint inhibitor therapy) within 4 weeks of the first planned dose of investigational therapy. Participants must have recovered from all adverse events (AEs) due to previous therapies to =< grade 1 or baseline. Participants with =< grade 2 neuropathy, alopecia, or other non-relevant AEs may be deemed eligible at the discretion of the principal investigator. If a participant received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting study treatment. Additionally, lenvatinib should not be administered for at least two weeks following major surgery and until adequate wound healing. Has received prior radiotherapy within 2 weeks of start of study intervention. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation (=< 2 weeks of radiotherapy) to noncentral nervous system (CNS) disease. Has received a live vaccine within 30 days prior to the first dose of study drug. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guerin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (e.g., FluMist) are live attenuated vaccines and are not allowed. Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to the first dose of study intervention. Participants who have entered the follow-up phase of an investigational study may participate as long as it has been 4 weeks after the last dose of the previous investigational agent. Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug. Has history of a second malignancy, unless potentially curative treatment has been completed with no evidence of malignancy for 2 years. The time requirement does not apply to participants who underwent successful definitive resection of basal cell carcinoma of the skin, squamous cell carcinoma of the skin, superficial bladder cancer, in situ cervical cancer, or other in-situ cancers. Has known active CNS metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to first dose of study intervention. Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is allowed. Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. Has an active infection requiring systemic therapy. Has a known history of human immunodeficiency virus (HIV) infection. Has a known history of chronic hepatitis B or hepatitis C virus infection. Has a known history of active Bacillus tuberculosis (TB) Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. Has had an allogenic tissue/solid organ transplant. Participants having greater than 1+ proteinuria on urinalysis will undergo 24-hour (h) urine collection for assessment of proteinuria. Participants with urine protein greater than or equal to 2 g/24-h will be ineligible. Has a left ventricular ejection fraction (LVEF) below the institutional (or local laboratory) normal range, as determined by multigated acquisition (MUGA) or echocardiogram (ECHO). Has clinically significant cardiovascular disease within 12 months from first dose of study intervention, including New York Heart Association (NYHA) class III or IV congestive heart failure, unstable angina, myocardial infarction, cerebral vascular event, or cardiac arrhythmia associated with hemodynamic instability. Note: medically controlled arrhythmia would be permitted. Prolongation of corrected QT using Fridericia's formula (QTcF) interval to > 480 ms Bleeding or thrombotic disorders or subjects at risk for severe hemorrhage. Subject with known deep vein thrombosis/pulmonary embolism that are under appropriate anti-coagulation treatment are eligible. Has radiographic evidence of tumor encasement or invasion of a major blood vessel, or intra-tumoral cavitation. Any history of hypertensive crisis with prior anti-VEGF therapy. Have pre-existing >= grade 3 gastrointestinal or non-gastrointestinal fistula