Prostate Specific Membrane Antigen (PSMA) or (FACBC) PET/CT Site-Directed Therapy for Treatment of Prostate Cancer, Flu-BLAST-PC Study

Inclusion Criteria: * Patient must have histologically or cytologically documented evidence of prostate adenocarcinoma * Patient must previously have undergone radical prostatectomy * Patient must previously have undergone either adjuvant or salvage radiation therapy to the prostatic fossa +/- whole pelvis * Patient must have a prostate specific antigen (PSA) \>= 0.2 and \< 10 ng/mL. If there is only one PSA value that has risen to \>= 0.2 with this biochemical recurrence, a second PSA value must be confirmed to be within \>= 0.2 and \< 10 ng/mL at least 2 weeks from the first value and within 28 days of enrollment * PSA doubling time must be calculated utilizing either all PSA measurements \> 0.1 ng/mL from most recent biochemically-recurred (BCR) or the most recent 3 PSA measurements \> 0.1 ng/mL (if the latter, all 3 PSA measurements must be \> 2 weeks apart to be used in the calculation). PSA doubling time must be \> 3 months and \< 18 months. The Memorial Sloan Kettering PSA doubling time calculator should be used * Patient must have no previous evidence of radiographically detectable metastatic prostate cancer by conventional CT and bone scan imaging * Patient must have total testosterone level \> 120 ng/dL demonstrated within 42 days of enrollment * Patient must have Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count (ANC) \>= 1.0 X 10\^9/L * Platelet count \>= 100 X 10\^9/L * Hemoglobin \>= 9 g/dL * Potassium \>= 3.5 * Serum bilirubin =\< 1.5 X upper limit of normal (ULN) or =\< 3 X ULN for patients with documented Gilbert's syndrome * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 X ULN * Creatinine clearance (Cr Cl) \>= 30 mL/min as estimated by the Cockcroft-Gault criteria or as determined by 24 hour Cr Cl measurement * Patient must be \>= 18 years of age on day of signing informed consent * Patient must be able to understand and authorize informed consent Exclusion Criteria: * Chronic active hepatitis B or C * History of a second, non-prostate malignancy that required systemic therapy in the last 2 years except cancer in situ of bladder and non-melanomatous cancers of the skin * Patient with a serious underlying medical condition that would otherwise impair the patient's ability to undergo fluciclovine or PSMA PET/CT imaging or receive subsequent treatment * Any condition that would alter the patient's mental status, prohibiting understanding and/or authorization of informed consent * Expected lifespan of less than 12 weeks * Inability to lay still for imaging * Weight \> 300 lbs. (due to equipment specifications) * Any other underlying medical condition that, in the opinion of the investigator, would impair the ability of the patient to receive or tolerate the planned treatment and/or follow up