This phase II trial studies how well prostate specific membrane antigen (PSMA) or fluciclovine positron emission tomography (PET)/computed tomography (CT) site-directed therapy works for treating patients with prostate cancer. PSMA or fluciclovine PET/CT may detect prostate cancer early and may help to show whether patients benefit from site directed treatment to PET detected abnormalities.

OUTLINE: Patients are assigned to 1 of 3 groups.

GROUP I: Patients for whom initial fluciclovine or PSMA PET/CT does not reveal any abnormalities outside the prostatic fossa undergo PSA rechecks every 3 months, and undergo fluciclovine or PSMA PET/CT once PSA is > 2 ng/ml. If still no abnormalities are found outside of the prostatic fossa, patients continue to undergo PSA rechecks every 3 months, and undergo fluciclovine or PSMA PET/CT once PSA is > 5 ng/ml. Patients are off study for treatment plan once PSA reaches 10 ng/ml.

GROUP II: Patients undergo fluciclovine or PSMA PET/CT and who have =< 3 regions of metastatic disease outside of the prostatic fossa that are amenable to metastasis-directed therapy undergo lymphadenectomy or radiation therapy. Six to ten weeks after surgery, patients receive abiraterone acetate 1000 mg orally (PO) once daily (QD) and prednisone PO QD. Treatment repeats every 4 weeks for 6 cycles in the absence of disease progression or unacceptable toxicity. Patients may start radiation therapy after 2 cycles of abiraterone acetate and prednisone.

GROUP III: Patients undergo fluciclovine or PSMA PET/CT and who have > 3 regions of metastatic disease receive abiraterone acetate and prednisone as in Group II.

After completion of study treatment, patients are followed up at 37, 49, 61, 73, 85, 97, 109, and 121 weeks, at 36, 42, 48, 54, 60, and 66 months, and then annually thereafter.

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