Pembrolizumab and Lenvatinib in Advanced/Metastatic Neuroendocrine Prostate Cancer

Inclusion Criteria: 1. Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. 2. Age ≥ 18 years at the time of consent. 3. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 7 days prior to the date of registration. 4. The subject has histologically proven prostate cancer with radiologic evidence of metastases and at least one of the following: * Small-cell or NEPC morphology (determined by the enrolling center) on the basis of tissue sample. * Prostate adenocarcinoma with greater than 50% IHC staining for neuroendocrine markers (e.g., chromogranin and synaptophysin). * Presence of visceral metastases or high volume disease (\> 4 sites of metastases) with a PSA of ≤ 5. * Serum chromogranin A level ≥ 5× upper limit of normal (ULN) and/or serum neuron specific enolase (NSE) ≥ 2× ULN. * RBI deletions or mutations noted on genomic testing. * Trans-differentiated carcinoma or poorly-differentiated carcinoma. 5. Subject has adequate organ function as defined in the table below; all screening labs to be obtained within 10 days prior to Cycle 1 Day 1. * Absolute neutrophil count (ANC) ≥ 1500/mm3without colony stimulating factor support * Platelets ≥ 100,000/mm3 * Hemoglobin ≥ 9 g/dL. Transfusions are allowed as needed. * Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 30 mL/min. For creatinine clearance estimation, the Cockcroft and Gault equation should be used. * Bilirubin ≤ 1.5 x the upper limit of normal (ULN) OR direct bilirubin ≤ULN for participants with total bilirubin levels \>1.5 × ULN. For subjects with known Gilbert's disease, bilirubin ≤ 3.0 mg/dL * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 2.5 x ULN if no liver involvement, or ≤ 5 and/or ULN with liver involvement * International normalized ratio (INR) OR prothrombin time (PT), Activated partial thromboplastin time (aPTT) ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants * Urine protein \< 2+ by urine dipstick 6. A male participant must agree to use of contraception during the treatment period and for at least 120 days after the last dose of study treatment and refrain from donating sperm during this period. 7. Projected life expectancy of at least 6 months as determined by treating physician. 8. As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. Exclusion Criteria: 1. Received prior therapy with VEGF-TKI, immune checkpoint inhibitor, an anti-PD-1, anti-PD-L1, or anti PD L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX 40, CD137). 2. Received prior systemic anti-cancer therapy including investigational agents within 3 weeks prior to registration. NOTE: Participants must have recovered from all AEs due to previous therapies to ≤ Grade 1 or baseline. Participants with ≤ Grade 2 neuropathy may be eligible. NOTE: If participant received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting study treatment. 3. Received more than two prior chemotherapy regimens for metastatic prostate cancer. Prior therapy with androgen receptor axis targeted agents is allowed but needs to be discontinued at least 2 weeks prior to study therapy. Prior therapy with Rad-223 or other radiopharmaceuticals is permitted but study therapy should be started at least 4 weeks after the last dose. 4. Concurrent treatment with anti-androgen medications. NOTE: LHRH agonists and GNRH antagonists may be continued. All oral anti androgens should be discontinued. 5. Received prior radiotherapy within 2 weeks of start of study treatment. 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 non-CNS disease. 6. Currently participating in or has participated in a study of an investigational agent or has used an investigational device within 3 weeks prior to the first dose of study treatment. NOTE: Participants who have entered the follow-up phase of an investigational study may participate as long as it has been 3 weeks after the last dose of the previous investigational agent. 7. Uncontrolled blood pressure (Systolic BP\>140 mmHg or diastolic BP \>90 mmHg) despite an optimized regimen of antihypertensive medication. 8. Presence of non-healing wounds after surgical procedures. 9. 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 treatment. 10. Received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of study drug. Administration of killed viruses are allowed. All COVID-19 vaccines are permitted at any time before or during the study. 11. 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, or cardiac arrhythmia requiring medical treatment at Screening. 12. Bleeding or thrombotic disorders or subjects at risk for severe hemorrhage. The degree of tumor invasion/infiltration of major blood vessels (e.g. carotid artery) should be considered because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following lenvatinib therapy. 13. Subjects having \> 1+ proteinuria on urine dipstick testing unless a 24-hour urine collection for quantitative assessment indicates that the urine protein is \<1 g/24 hours. 14. 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. 15. Severe hypersensitivity (≥ Grade 3) to pembrolizumab and/or any of its excipients. 16. Severe hypersensitivity (≥ Grade 3) to lenvatinib and/or any of its excipients. 17. 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 (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. Replacement steroids for adrenal insufficiency or daily dose equivalent of 10 mg prednisone are allowed 18. History of severe (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. 19. Active uncontrolled infection. 20. Known additional malignancy that is progressing or has required active treatment within the past 3 years. NOTE: Participants with basal cell carcinoma of the skin, squamous cell carcinoma of the skin or carcinoma in situ (e.g. breast carcinoma, cervical cancer in situ) that have undergone potentially curative therapy are not excluded. Subjects with other solid tumors treated curatively and without evidence of recurrence for at least 2 years prior to enrollment may be eligible for study after discussion with the sponsor-investigator. 21. Known history of Human Immunodeficiency Virus (HIV). NOTE: HIV testing is not required unless mandated by a local health authority. 22. 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. NOTE: Hepatitis B and Hepatitis C testing is not required unless clinical history indicates that this is likely. 23. Known history of active TB (Bacillus Tuberculosis).