Inclusion Criteria:  To be eligible for this trial, patients must meet all of the following eligibility criteria.  Patients with histologically confirmed locally advanced or metastatic endometrial carcinoma that is not amenable to curative surgical- or radiation-based intervention, who have received or declined at least one-line systemic chemotherapy. Known microsatellite stable (MSS) as pre-identified in a Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory At least one measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Age ≥18 years. Women of child-bearing potential (WOCBP) must have a negative serum pregnancy test prior to administration of the first dose of futibatinib within 7 days prior to initiation of therapy (C1D1), and must agree to use effective birth control initiated immediately following negative serum pregnancy test during screening period, during the study, and for at least 180 days after the last dose (or longer based on local requirements). Female patients are not considered to be of child-bearing potential if they are post-menopausal (no menses for 12 months without an alternative medical cause) or permanently sterile (hysterectomy, bilateral salpingectomy, or bilateral oophorectomy).  Adequate organ functions as defined below:  Absolute neutrophil count (ANC) ≥ 1,000 /μL. Hemoglobin (Hb) ≥ 9 g/dL. Platelets ≥ 75,000 /μL. Total bilirubin ≤ 1.5 x ULN (upper limit of normal); or total bilirubin < 3 x ULN with direct bilirubin ≤ ULN in patients with well documented Gilbert's Syndrome.  ALT ≤ 3 x ULN or ≤ 5 x ULN if liver metastases persist. Serum phosphate ≤ 1.5 x ULN. Serum calcium ≤ ULN. Serum albumin ≥ 3 g/dL. Serum creatinine ≤ 1.5 x ULN or calculated creatinine clearance (CrCl) ≥ 40 mL/min by the Cockcroft-Gault method* or 24-hour urine collection.  *CrCl = (140-age) x (weight [kg]) x 0.85 / (72 x serum creatinine mg/dL)  Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1. No prior treatment with anti-PD-(L)1 therapy and/or a FGFR inhibitor. Ability to read and fully understand the requirements of the trial, willingness to comply with all trial visits and assessments, and willingness and ability to sign an institutional review board (IRB)-approved written informed consent document. Any prior radiation must have been completed at least 14 days prior to the start of study drugs, and patients must have recovered from any acute adverse effects prior to the start of study treatment (Radiotherapy for extended field within 4 weeks or limited field radiotherapy within 2 weeks). Ability to take oral medications without medical history of malabsorption or other chronic gastrointestinal disease, or other conditions that may hamper compliance and/or absorption of the study agents (feeding tube is not permitted). Provision of an archival tissue block, or 10 formalin-fixed paraffin-embedded (FFPE) slides obtained within 6 months prior to study entry; or agreeing to have biopsies if archival tissues are not available.  Exclusion Criteria:  Patients who meet any of the following criteria will be not eligible for the study: Uncontrolled intercurrent illness including but not limited to ongoing or active infection requiring intravenous antibiotics, symptomatic congestive heart failure (New York Heart Association Class III or IV), history of myocardial infarction, unstable angina, stroke or transient ischemic attack within 6 months before study enrollment, history or current evidence of uncontrolled ventricular arrhythmia. Congenital long QT syndrome, or any known history of torsade de pointes, or family history of unexplained sudden death. Chronic diarrhea diseases considered to be clinically significant in the opinion of the Investigator. Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or futibatinib and pembrolizumab administration, or may interfere with the interpretation of study results, and in the judgment of the Investigator would make the patient inappropriate for entry into this study. History and/or current evidence of any of the following disorders: Non-tumor related alteration of the calcium-phosphorus homeostasis that is considered clinically significant in the opinion of the Investigator; Ectopic mineralization/calcification, including but not limited to soft tissue, kidneys, intestine, or myocardia and lung, considered clinically significant in the opinion of the Investigator; and Retinal or corneal disorder confirmed by retinal/corneal examination and considered clinically significant in the opinion of the Investigator and a trained ophthalmologist who performs the test. Having not recovered from a major surgical procedure or significant traumatic injury (i.e., still needing additional surgical or medical care for these issues): major surgical procedures ≤ 28 days of treatment entry, or minor surgical procedures ≤ 7 days. No waiting period required following port-a-cath or other central venous access placement. Unresolved clinically significant Grade 2 toxicity from prior therapy. Patient has an inability to swallow oral medications. Note: Patient may not have a percutaneous endoscopic gastrostomy (PEG) tube or be receiving total parenteral nutrition (TPN). Clinically active bleeding, or active gastric or duodenal ulcer. Fridericia's corrected QT interval (QTcF =QT/∛(60/HR) ) > 470 ms on ECG conducted during Screening. Patients with an atrioventricular pacemaker or other condition (for example, right bundle branch block) that renders the QT measurement invalid are an exception and the criterion does not apply. History of allergic reactions to the study drugs, or any component of the products. Currently receiving an investigational drug in a clinical trial or participating in any other type of medical research judged not to be scientifically or medically compatible with this study. If a patient is currently enrolled in a clinical trial involving non-approved use of a device, then agreement with the investigator and the sponsor is required to establish eligibility. Any treatment specifically for systemic tumor control given within 3 weeks before the initiation of the study drugs, within 2 weeks if cytotoxic agents were given weekly, within 6 weeks for nitrosoureas or mitomycin C, within 5 half-lives for targeted agents with half-lives and pharmacodynamic effects lasting < 5 days, or failure to recover from toxic effects of any therapy before the initiation of the study drugs. A drug that has not received regulatory approval for any indication within 14 or 21 days of treatment for a non-myelosuppressive or myelosuppressive agent, respectively. 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: Bacille Calmette-Guérin vaccine, measles, mumps, rabies, rubella, typhoid vaccine, varicella/zoster, and yellow fever. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines are live attenuated vaccines and are not allowed. Received strong inhibitors and inducers and sensitive substrates of CYP3A4 within 2 weeks (see appendix A). Symptomatic primary tumors or metastasis in the brain and/or central nervous system that are uncontrolled with antiepileptics and require steroids at a dose of prednisone > 10 mg/day or equivalent. Evidence of leptomeningeal or lymphangitic carcinomatosis. A history of another primary malignancy that is currently clinically significant, requiring active intervention except for hormone therapy. Concurrent immunosuppressive therapy or steroid (> 10 mg/day prednisone or equivalent). Previously documented or suspected autoimmune disease requiring active immunosuppressive therapy (adequately treated skin rash or replacement therapy for endocrinopathies is not excluded), and history of or current pneumonitis. Lactation or pregnancy. Known human immunodeficiency virus requiring HAART treatment due to unknown drug-drug interactions or known active hepatitis B or C virus infection.