Finite Androgen Ablation With or Without Abiraterone Acetate and Prednisone in Treating Patients With Recurrent Prostate Cancer

Inclusion Criteria: * Have signed an informed consent document indicating that the subjects understand the purpose of and procedures required for the study and are willing to participate in the study * Written Authorization for Use and Release of Health and Research Study Information has been obtained * Be willing/able to adhere to the prohibitions and restrictions specified in this protocol * Life expectancy \>= 12 months * ECOG performance status (PS) =\< 2 * Histologically documented diagnosis of adenocarcinoma of the prostate (PCa) with no histologic variants * Prostate cancer recurrence after definitive local therapy (radical prostatectomy and/or radiation therapy) as evidenced by rising serum PSA, without evidence of metastases by bone scan or computed tomography (CT) scan * After radiation: A rising PSA taken to indicate recurrent prostate cancer in patients with previous definitive external beam radiotherapy will be defined as PSA of 1.0 * After Radical Prostatectomy: A rising PSA taken to indicate recurrent prostate cancer in patients with previous radical prostatectomy will be defined by the criteria of the American Urological Association as any PSA measurement of 0.2, with a subsequent measurement \> 0.2 ng/mL * Patients who have received androgen ablative therapy for less than 8 weeks immediately prior to initiation of study drug are eligible provided they had only PSA evidence of progression (as defined above) with no visible metastases by CT-scan and bone …