Temsirolimus in Treating Patients With Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Diagnosis of multiple myeloma (MM) * Salmon-Durie stage IIA or IIIA OR progressive stage IA disease * Meets at least 1 major AND 1 minor criterion OR at least 3 minor criteria * The following are considered major criteria: * Plasmacytoma on tissue biopsy * Bone marrow plasmacytosis with \>= 30% plasma cells * Monoclonal globulin spike on serum protein electrophoresis exceeding 3.5 g/dL for immunoglobulin (Ig) G peaks or 2.0 g/dL for IgA peaks OR the presence of Bence-Jones protein of \>= 1 g/24 hour-urine collection * The following are considered minor criteria: * Bone marrow plasmacytosis 10-29% * Monoclonal globulin spike present, but less than the levels defined for a major criterion * Lytic bone lesion * Decrease in normal IgM \< 50 mg/dL, IgA \< 100 mg/dL, or IgG \< 600 mg/dL * No non-secretory MM (absent serum or urinary M-protein) * Failed at least 1 prior systemic therapy\* (e.g., chemotherapy, high-dose corticosteroids, thalidomide, or bortezomib) for the treatment of MM * No solitary plasmacytoma * Performance status - ECOG 0-2 * More than 6 months * Absolute neutrophil count \> 1,200/mm\^3 * Platelet count \> 75,000/mm\^3 * AST and ALT =\< 2.5 times upper limit of normal (ULN) * Bilirubin =\< 1.5 times ULN * Creatinine =\< 1.5 times ULN * No symptomatic congestive heart failure * No unstable angina pectoris * No cardiac arrhythmia * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * Fasting cholesterol =\< 350 mg/dL * Triglycerides =\< 400 mg/dL * No other concurrent uncontrolled illness * No active or ongoing infection requiring oral or IV antibiotics * No prior allergic reaction to compounds of similar chemical or biological composition to CCI-779 * No other prior or concurrent malignancy or myelodysplasia except for the following: * Basal cell or squamous cell skin cancer * Carcinoma in situ of the cervix * Localized cancer treated with surgery only with no evidence of disease for \> 5 years * No psychiatric illness or social situation that would preclude study compliance * More than 4 weeks since prior thalidomide and recovered * Prior high-dose chemotherapy and stem cell transplantation allowed * More than 4 weeks since prior chemotherapy and recovered * More than 4 weeks since prior high-dose corticosteroids and recovered * More than 4 weeks since prior bortezomib and recovered * More than 4 weeks since other prior anti-myeloma systemic therapy and recovered * No concurrent combination antiretroviral therapy for HIV-positive patients * No other concurrent investigational agents * No other concurrent anticancer therapy