Hormone Therapy and Temsirolimus in Treating Patients With Relapsed Prostate Cancer

Inclusion Criteria: * All patients must sign an informed consent indicating that they are aware of the investigational nature of this study; patients must also have signed an authorization for the release of their protected health information * Patients must have histologically confirmed adenocarcinoma of the prostate recurring after local therapy (radical prostatectomy and/or radiation therapy) as evidenced by rising serum PSA * Prostate-Specific Antigen (PSA) Doubling Time (PSADT) =\< 12 months after local therapy (prostatectomy and/or definitive radiation) as determined by linear regression of all available PSA values within 6 months of initiation of androgen ablation (for patients who underwent prostatectomy, at least one PSA measurement of \>= 1.0 ng/mL; for patients who underwent radiation, at least one PSA measurement of \>= 3.0 ng/mL and \>= 150% postradiation nadir) * No evidence of metastasis as determined by bone scan or computed tomography (CT) scan * Initiation of Androgen Ablation of less than 8 weeks' duration prior to study entry is permitted * Leukocytes ≥ 3,000/mcl * Absolute neutrophil count ≥ 1,000/mcl * Hemoglobin ≥ 8.0g/dl * Eligibility level for hemoglobin may be reached by transfusion * Platelet count \>= 100,000/μL * Total bilirubin ≤1.5 X laboratory ULN * AST and/or ALT ≤ 3 X laboratory ULN * Creatinine ≤ 1.5 X laboratory ULN OR calculated creatinine clearance ≥ 60 ml/min/1.73 m\^2 for patients w/creatinine levels above the laboratory ULN * Serum cholesterol level \< 350 mg/dl * Triglyceride level \< 300mg/dl * ECOG performance status 0, 1 or 2 * The effects of Temsirolimus on the developing human fetus are unknown; for this reason men must agree to use contraception from the time of study enrollment continuing for the duration of study participation * Patients must be registered in the MDACC institutional database prior to treatment with study drug * PSA \< 40 ng/ml Exclusion Criteria: * Patients with histologic variants other than adenocarcinoma in the primary tumor * Patients may not be receiving any other investigational agents * Patients may not be receiving concomitant immunotherapy or immunosuppressive therapy * Patients may not have received prior systemic treatment for prostate cancer (other than no more than 3 months of prior treatment with androgen ablation in neoadjuvant and/or adjuvant setting and at least a year must have elapsed since last administration) unless initiation of Androgen Ablation of less than 8 weeks' duration prior to study entry is permitted * Patient with uncontrolled intercurrent illness including, but not limited to ongoing or active infection requiring parenteral therapy on day 1 of protocol treatment, symptomatic congestive heart failure resulting in a resting O2 saturation of \< 92% on room air, unstable angina pectoris, myocardial infarction within the previous 6 months, or use of ongoing maintenance therapy for life-threatening ventricular arrhythmia, known pulmonary hypertension or pneumonitis * Patients in a severely compromised immunological state, including being positive for the human immunodeficiency virus (HIV) due to possible pharmacokinetic interactions with HAART therapy * Patients diagnosed with acute or chronic hepatitis B or C * Patients using immunosuppressive agents, including intravenous corticosteroids, within 3 weeks of study entry * Patients must not have a history of any other cancer (except nonmelanoma skin cancer), unless in complete remission and off of all therapy for that disease for a minimum of 3 years