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 scan (within 6 weeks) prior to starting androgen ablation * White blood cell (WBC) \>= 3.5 x 10\^9/L * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Hemoglobin (Hb) \>= 9.0 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 1.5 x the upper limit of normal * Serum potassium of \>= 3.5 mEq/L * Serum albumin of \>= 3.0 g/dL * Serum creatinine =\< 1.5 x ULN * Patients must have recovered from prior treatment regimens, e.g. surgery, radiation * A patient who is sexually active and their partner must agree and use two reliable barrier forms of contraception (for example, condoms and diaphragm), from first day of study drug administration until for 1 week after last dose of abiraterone acetate, unless partner is post-menopausal * Able to swallow the study drug whole as a tablet * Willing to take abiraterone acetate on an empty stomach; no food should be consumed at least two hours before and for at least one hour after the dose of abiraterone acetate is taken Exclusion Criteria: * Patients who have received prior hormonal therapy are excluded from the trial, except for: patients who have received up to 6 months of hormonal therapy as neoadjuvant therapy before radical prostatectomy or while on radiation therapy, as long as more than 1 year has elapsed between discontinuation of the neoadjuvant hormonal therapy and initiation of hormonal treatment for relapsing disease * Any known metastases * Prolonged corrected QT (QTc) interval on pre-entry electrocardiogram (\>= 450 msec) * Clinically significant heart disease as evidenced by myocardial infarction, or arterial thrombotic events in the past 6 months, severe or unstable angina, or New York Heart Association (NYHA) Class II-IV heart disease or cardiac ejection fraction measurement of \< 50% at baseline * Significant co-morbidity that could affect the safety or evaluability of participants as assessed by the treating physician and or principal investigator * Prior therapy with strontium-89, samarium, rhenium-186 etidronate, chemotherapy or androgen biosynthesis inhibitors for prostate cancer is not allowed. Previous immunologic, homeopathic, natural, or alternative medicine therapies are acceptable provided treatment ended greater than 28 days prior to initiation of study drug * Patients who, in the opinion of the investigator, are unable to comply with the requirements of the study protocol are not eligible * Active infection or other medical condition that would make prednisone/prednisolone (corticosteroid) use contraindicated * Active or symptomatic viral hepatitis * History of pituitary or adrenal dysfunction * Administration of an investigational therapeutic drug within 30 days of cycle 1 day 1 * Have known allergies, hypersensitivity, or intolerance to abiraterone acetate or prednisone or their excipients * Have a history of gastrointestinal disorders (medical disorders or extensive surgery) that may interfere with the absorption of the study agents * Have a pre-existing condition that warrants long-term corticosteroid use in excess of study dose