Fecal Microbiota Transplant and Pembrolizumab for Men With Metastatic Castration Resistant Prostate Cancer.

Inclusion Criteria: 1. Be willing and able to provide written informed consent/assent for the trial prior to the performance of any protocol-related procedures that are not part of normal care. 2. Be ≥ 18 years of age at the time the informed consent is signed. 3. Histologically or cytologically documented adenocarcinoma of the prostate. Patients without histologically confirmed adenocarcinoma may be eligible if both the treating physician and the study PI agree that the patient's history is unambiguously indicative of advanced adenocarcinoma. 4. Receive care through a Veterans Affairs Hospital or is eligible for care at VHA. 5. Have metastatic castration resistant prostate cancer with castrate-level testosterone (\<50 ng/dL). a. Participants must maintain a castrate-level testosterone during the study. 6. Have disease progression defined by one or more of the following three criteria: 1. PSA \> 2.0 ng/mL at time of screening and rising PSA by at least 2 consecutive measurements a minimum of 1-week apart. 2. Soft tissue progression as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1. 3. Bone disease progression as defined by the PCWG3. 7. Have measurable disease based on RECIST v.1.1 modified as described by the PCWG3 or non-measurable disease with bone metastases. a. Participants with measurable disease must have at least one lesion that is ≥10 mm in longest diameter for a soft tissue lesion, or ≥15 mm in short axis for a lymph node. 8. Have a metastatic lesion that can be safely biopsied and be willing to undergo the tumor biopsy. If the participant is on anticoagulation, it must be safe to hold the anticoagulation for the biopsy. 9. Have had a PSA response to enzalutamide (defined as a PSA decline of 50% or more), but now showing signs of PSA and/or radiographic progression per PCWG3 and/or RECIST 1.1. (Patients must be continuously on enzalutamide prior to joining the main study. They may not have had progression on enzalutamide and then challenge with new therapy and then restarted on enzalutamide.) 10. Participants must discontinue antiandrogen therapy (ie, bicalutamide, flutamide, nilutamide) at least 4-6 weeks prior to registration with no evidence of PSA decline after washout. 1. Bicalutamide: Washout period at least 6 weeks 2. Flutamide and nilutamide: Washout period at least 4 weeks 11. Participants must discontinue therapies for mCRPC, with the exception of enzalutamide and a GnRH agent, for 5 half-lives or 28 days, whichever is shorter. 1. Prior treatment with sipuleucel-T, radium-223, or abiraterone is allowed. 2. Tissue biopsy may be performed during washout period. 12. Have a performance status of 0 or 1 according to the Eastern Cooperative Oncology Group (ECOG) scale. 13. Demonstrate adequate organ function on screening laboratory tests performed within 14 days of treatment initiation and as evidenced by: 1. Hemoglobin ≥ 9.0 g/dL or ≥ 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within ≤ 7 days of assessment) 2. WBC \> 2,000/mm3 without growth factor support (within ≤ 7 days days of assessment) 3. Absolute neutrophil count ≥ 1,500/mm3 without growth factor support or transfusion (within \< 28 days of assessment) 4. Platelet count ≥ 100,000/mm3 5. Serum creatinine \< 1.5 x upper limit of normal (ULN) or measured or calculated (calculated per institutional standard) creatinine clearance ≥ 60 mL/min for participant with creatinine levels \> 1.5 x institutional ULN. GFR can also be used in place of creatinine or CrCl. 6. Serum total bilirubin \< 1.5 x ULN or ≤ 2.0 x ULN for participants with liver metastases * Participants with Gilbert's syndrome must have ≤ 3 x ULN and no liver lesions 7. Aspartate transaminase (AST) (SGOT) and alanine transaminase (ALT) (SGPT) ≤ 2.5 x ULN or ≤ 5.0 x ULN for participants with liver metastases. 8. Albumin ≥ 2.5 mg/dL. 9. International normalized ratio (INR) or prothrombin time (PT) ≤ 1.5 x ULN unless participant is receiving anticoagulant therapy, as long as PT is within therapeutic range of intended use of anticoagulants. 10. Activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN unless participant is receiving anticoagulant therapy, as long as PTT is within therapeutic range of intended use of anticoagulants. 14. Male participants of reproductive potential must agree to use an adequate method of contraception, starting with the first dose of study intervention and through 120 days after the last dose of study therapy. Contraception is not required if the patient's partner is a woman who is post-menopausal. Subjects of reproductive potential must agree to avoid impregnating a partner by complying with one of the following: Practice abstinence from heterosexual activity; abstinence (relative to heterosexual activity) can be used as the sole method of contraception if it is consistently employed as the subject's preferred and usual lifestyle. Periodic abstinence (e.g., calendar, ovulation, sympto-thermal, post-ovulation methods, etc.) and withdrawal are not acceptable methods of contraception. OR Have their partner use acceptable contraception during heterosexual activity. Acceptable methods of contraception are: Single method (one of the following is acceptable): * Intrauterine device (IUD) * Contraceptive rod implanted into the skin. Combination method (requires use of two of the following): * Diaphragm with spermicide (cannot be used in conjunction with cervical cap/spermicide) * Cervical cap with spermicide (nulliparous women only) * Contraceptive sponge (nulliparous women only) * Male condom or female condom (cannot be used together) * Hormonal contraceptive: oral contraceptive pill (estrogen/progestin pill or progestin-only pill), contraceptive skin patch, vaginal contraceptive ring, or subcutaneous contraceptive injection. 15. Participants must be able and willing to comply with the study visit schedule and study procedures. Exclusion Criteria: 1. 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. Superficial bladder cancer is also permitted provided it is monitored and treated locally. 2. Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least 4 weeks prior to the first dose of study intervention 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 study intervention. This exception does not include carcinomatous meningitis, which is excluded regardless of clinical stability. 3. Uncontrolled disease-related bone pain or other symptoms that suggest the participant should imminently go onto chemotherapy. 4. Those with tumors having known microsatellite instability will be excluded. Participants found to have microsatellite instability in their tumors after analysis of the on-study biopsy will not be eligible to serve as FMT donors, but will be permitted on the study. 5. Prior taxane-based chemotherapy (in any setting- castration sensitive or resistant). 6. Has had prior therapy with an anti-CTLA4, anti-PD-1 or anti-PD-L1 antibody. 7. Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to the first dose of study treatment or who has not recovered (ie, ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. 8. Has 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. 9. Has had prior targeted small molecule therapy within 2 weeks prior to the first dose of study treatment or who has not recovered (ie, ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent. 1. Note: Participants with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. 2. Note: If a participant received major surgery, he must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. 10. 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. 11. Has received broad-spectrum antibiotics within 3 months of first dose of pembrolizumab. 12. Has a history of seizure, unless he had a mass in his brain that has been removed, such as a meningioma. 13. Has a diagnosis of immunodeficiency or conditions that need systemic corticosteroid replacement therapy \> 10 mg/day prednisone (or equivalent) or other immunosuppressive medications within 28 days prior to the first dose of study intervention. Inhaled steroids are permitted if necessary. 14. Has any active known or suspected autoimmune disease. Participants with vitiligo, type I diabetes mellitus, controlled autoimmune hypothyroidism, psoriasis not requiring systemic treatment, or other conditions under control are permitted to enroll. Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy \< 10 mg of prednisone/day for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. 15. Has a known history of active TB (Bacillus Tuberculosis). 16. Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis . 17. Has an active infection requiring systemic therapy. 18. Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the participant's participation for the full duration of the trial, or is not in the best interest of the participant to participate, in the opinion of the treating investigator. 19. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. 20. Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients Testing positive for any of the following infectious diseases (criteria 21-22). 21. Evidence of the following infectious agents: 1. Stool pathogens: Clostridium difficilie toxin B by PCR, Giardia antigen, Cryptosporidium antigen, Acid-fast stan for Cyclospora or Isospora, Ova and parasites, Helicobacterial pylori antigen positivity, Salmonella spp., Shigella spp., Campylobacter spp., Shiga-toxin producing Escherichia coli, Methicillin Resistant Staphylococcus aureus, Vancomycin Resistant Enterococcus spp., Carbapenem Resistant Enterobacteriaceae, ESBL producing E. coli, Aeromonas spp., Plesiomonas spp., Yersinia spp., Vibrio spp., Entamoeba histolytica, Rotavirus, Adenovirus, Norovirus 2. Blood pathogens: Positive for HIV, type 1 or 2; Hepatitis A IgM; Hepatitis B antigen, anti-HBC (both IgG and IgM), and anti-HBs; Hepatitis C antigen; T. pallidum antibody; FTA-ABS (if positive T. pallidum screen) 3. COVID-19 (unless vaccinated against COVID-19) 22. Risk factors for contracting an illness: 1. Ongoing high-risk behaviors (e.g. men who have sex with men, men who have sex for money, men who use intravenous drugs) 2. Tattoo or body piercing within 6 months 3. Risk factors for Creutzfeldt-Jakob disease 4. Travel within the past 6 months to areas of the world where diarrheal illnesses are endemic or risk of traveler's diarrhea is high 23. Known current communicable disease (e.g. upper respiratory infection). 24. Gastrointestinal co-morbidities: history of inflammatory bowel disease, history of irritable bowel syndrome or idiopathic chronic constipation, history of chronic diarrhea related to inflammatory bowel disease with current diarrheal symptoms, history of gastrointestinal malignancy or known polyposis. 25. Major immunosuppressive medications, e.g., calcineurin inhibitors, exogenous glucocorticoids, biologic agents, etc. 26. Systemic antineoplastic agents other than enzalutamide and LHRH agonist or antagonist in the past 4 weeks. 27. Has received a live vaccine within 30 days of planned start of study intervention. Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (eg, Flu-Mist®) are live attenuated vaccines, and are not allowed.