A Phase II a Study of Lenvatinib, a Multi-targeted Tyrosine Kinase Inhibitor, Combined With Pembrolizumab (PD-1 Inhibitor) for the Treatment of Metastatic Gastroesophageal Cancer Patients Who Have Progressed on First or Subsequent Line Therapies

Inclusion Criteria: * Be willing and able to provide written informed consent/assent for the trial. * Be greater than 18 years of age on day of signing informed consent. * Have histologically or cytologically confirmed metastatic or recurrent gastric or GEJ adenocarcinoma. * Have measurable disease based on RECIST 1.1. * Must have received and have progressed, or are intolerant to at least one systemic regimen (platinum- or fluoropyrimidine-based chemotherapy regimen for metastatic or recurrent disease.) * If Her2 positive, must have received and have progressed or are intolerant to treatment with trastuzumab for metastatic or recurrent disease. * Subjects must consent to provide archival tumor tissue (initial and subsequent tumor biopsy samples, if possible) for correlative biomarker studies. * Have a performance status of 0 or 1 on the ECOG Performance Scale. * Adequately controlled blood pressure with or without antihypertensive medications defined as BP \< 140/90 mmHg at screening and no change in antihypertensive mediation within 1 week prior to the Screening Visit. * Demonstrate adequate organ function as defined in Table 5, all screening labs should be performed within 10 days of treatment initiation. * Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * Female subjects of childbearing potential (Section 5.5.2) must be willing to use an adequate method of contraception as outlined in Section 5.5.2 - Contraception, for the course of the study through 120 days after the last dose of study medication. * Male subjects of childbearing potential (Section 5.5.2) must agree to use an adequate method of contraception as outlined in Section 5.5.2 - Contraception, starting with the first dose of study therapy through 120 days after the last dose of study therapy. Exclusion Criteria: * Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy (in dosing exceeding 10mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Has a known history of active TB (Bacillus Tuberculosis) * Hypersensitivity to pembrolizumab or lenvatinib or any of its excipients. * Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent. * Note: Subjects with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. * Note: If subject received major surgery, a minimum of four weeks must have passed and they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer. * Subjects having \>1+ proteinuria on urine dipstick testing will undergo 24h urine collection for quantitative assessment of proteinuria. Subjects with urine protein \>1g/24h will be ineligible. * Gastrointestinal malabsorption or any other condition in the opinion of the investigator that might affect the absorption of lenvatinib. * Prolongation of QTcF interval to \>480ms * Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) Class II, unstable angina, myocardial infarction or stroke within 6 months of the first dose of study drug. * Bleeding disorders or active significant bleeding (i.e. hemoptysis, hematochezia, hematemesis) within 3 weeks. * Thrombotic disorders or use of anticoagulants, such as warfarin, requiring therapeutic international normalized ratio (INR) monitoring. (treatment with low molecular weight heparin (LMWH) or direct acting oral anti-coagulants is allowed.) * History of prior gastrointestinal fistula and/or perforation. * Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. * 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 not considered a form of systemic treatment and is allowed. * Has known history of, or any evidence of active, non-infectious pneumonitis. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, 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. * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the screening visit through 120 days after the last dose of trial treatment. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent. * Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). * Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). * Has received a live vaccine within 30 days of planned start of study therapy. * Arrhythmias requiring Class Ia and III antiarrhythmics and/or grade \>2 bradycardia.