A Pilot Presurgical Trial of REGN5678 (Anti-PSMA x CD28) in Patients With High-risk, Localized Prostate Cancer Followed by Radical Prostatectomy

Inclusion Criteria: * Men ≥ 18 years of age * Histologically documented Gleason 8 or greater prostatic adenocarcinoma in at least 3 biopsy cores and at least 8 mm of disease on a single core of Gleason 8 or greater. Prostate biopsy within 3 months of screening is allowed for entry requirements. Prostate biopsy must be reviewed at MD Anderson Cancer Center. Patients with small cell, neuroendocrine, or transitional cell carcinomas or mixed histologies are not eligible * Eastern Cooperative Oncology Group (ECOG) performance status (PS) grade of 0 or 1 * No evidence of metastatic disease as documented by technetium-99m (99mTc) bone scan and by computed tomography (CT) or magnetic resonance imaging (MRI) scans. Imaging may be obtained up to 60 days prior to enrollment * Localized or locally advanced disease deemed by the surgeon to be resectable. Patients must be appropriate candidates for radical prostatectomy plus pelvic lymph node dissection * No prior treatment for prostate cancer including prior surgery (excluding transurethral resection of the prostate \[TURP\]), cryoablation, pelvic lymph node dissection, radiation therapy, hormonal therapy or chemotherapy * Hemoglobin ≥ 11 g/dL * Absolute neutrophil count ≥ 1.5 x 10\^9/L * Platelet count ≥ 100 x 10\^9/L * Serum creatinine ≤ 1.5 x upper limit of normal (ULN) or estimated glomerular filtration rate \> 50 mL/min/1.73 m\^2. A 24-hour urine creatinine collection may substitute for the calculated creatinine clearance to meet eligibility criteria * Total bilirubin ≤ 1.5 x ULN * NOTES: Patients with Gilbert's syndrome do not need to meet total bilirubin requirements provided their total bilirubin is not greater than their historical level. Gilbert's syndrome must be documented appropriately as past medical history * Aspartate aminotransferase (AST) ≤ 2.5 x ULN * Alanine aminotransferase (ALT) ≤ 2.5 x ULN * Alkaline phosphatase (ALP) ≤ 2.5 x ULN * Consent to MD Anderson laboratory protocol PA13-0291 * Willing and able to comply with clinic visits and study-related procedures * Provide informed consent signed by study patient * To avoid risk of drug exposure through the ejaculate (even men with vasectomies), subjects must use a condom during sexual activity while on study drug and for 3 months following the last dose of study drug. If the subject is engaged in sexual activity with a woman of childbearing potential, a condom is required along with another effective contraceptive method consistent with local regulations regarding the use of birth control methods for subjects participating in clinical studies and their partners. Donation of sperm is not allowed while on study drug and for 3 months following the last dose of study drug Exclusion Criteria: * Prior hormone therapy for prostate cancer including orchiectomy, antiandrogens, ketoconazole, or estrogens (5-alpha reductase inhibitors allowed), or luteinizing hormone-releasing hormone (LHRH) agonists/antagonists * Currently enrolled in another interventional study * Concurrent treatment with systemic corticosteroids (prednisone dose \> 10 mg per day or equivalent) or other immunosuppressive drugs \< 14 days prior to treatment initiation. Steroids that are topical, inhaled, nasal (spray), or ophthalmic solution are permitted * History of or known or suspected autoimmune disease (exception\[s\]: subjects with vitiligo, resolved childhood atopic dermatitis, hypothyroidism, or hyperthyroidism that is clinically euthyroid at screening are allowed) * Known evidence of an active infection requiring systemic therapy such as human immunodeficiency virus (HIV), active hepatitis, or fungal infection. Patients with known HIV infection which is well-controlled (undetectable viral load by HIV ribonuclecid acid \[RNA\] polymerase chain reaction \[PCR\]) and CD4 counts greater than 350 are permitted to participate * History of clinically significant cardiovascular disease including, but not limited to: * Myocardial infarction or unstable angina ≤ 6 months prior to treatment initiation * Clinically significant cardiac arrhythmia * Deep vein thrombosis, pulmonary embolism, stroke ≤ 6 months prior to treatment initiation * Congestive heart failure (New York Heart Association class III-IV) * Pericarditis/clinically significant pericardial effusion * Myocarditis * Endocarditis * History of major implant(s) or device(s), including but not limited to: * Prosthetic heart valve(s) * Artificial joints and prosthetics placed ≤ 12 months prior to treatment initiation * Current or prior history of infection or other clinically significant adverse event associated with an exogenous implant or device that cannot be removed * Other prior malignancy (exceptions: adequately treated basal cell or squamous cell skin cancer, superficial bladder cancer, or any other cancer in situ currently in complete remission) ≤ 2 years prior to enrollment * Has received major surgery within 14 days of first administration of study drug * Encephalitis, meningitis, neurodegenerative disease (with the exception of mild dementia that does not interfere with activities of daily living \[ADLs\]) or uncontrolled seizures in the year prior to first dose of study therapy * Known history of, or any evidence of interstitial lung disease, or active, non-infectious pneumonitis (past 5 years) * Receipt of a live vaccine within 4 weeks of planned start of study medication * Prior allogeneic stem cell transplantation or recipients of organ transplants at any time, or autologous stem cell transplantation within 12 weeks of the start of study treatment * Any medical, psychological or social condition that in the opinion of the investigator, would preclude participation in this study