Vaccine Therapy With or Without Sirolimus in Treating Patients With NY-ESO-1 Expressing Solid Tumors

Inclusion Criteria: * Patients with any solid tumors at high risk of recurrence or with minimal residual disease; there may or may not be measurable or symptomatic disease (i.e., patients with bladder, brain, breast, esophageal, gastrointestinal, hepatocellular, kidney, lungs, melanoma, ovarian, prostate, sarcomas, and uterine) * Cancer types: * Prostate cancer: patients with metastatic, castrate refractory prostate cancer; the use of luteinizing hormone-releasing hormone (LHRH) agonist is allowed * Kidney cancer: patients with metastatic kidney cancer; prior therapies with cytokines, vascular endothelial growth factor (VEGF) and mechanistic target of rapamycin (serine/threonine kinase) (mTOR) inhibitors are allowed * Bladder cancer: patients with metastatic urothelial carcinoma; prior cisplatin-based therapies are allowed * Ovarian cancer: eligible patients may have asymptomatic residual measurable disease on physical examination and/or computed tomography (CT) scan, and/or may have an elevated cancer antigen (CA)-125; or may be in complete clinical remission after treatment for primary or recurrent disease * Brain tumors: histologic proof of one of the following: glioblastoma multiforme, anaplastic astrocytoma, anaplastic oligodendroglioma or anaplastic mixed glioma or anaplastic oligoastrocytoma; patients who have had recent cranial surgery are eligible for inclusion, but the vaccine may not be administered prior to postoperative day 14 * Uterine cancer: patients with advanced (stages II-IV) or recurrent disease who have completed standard therapy, currently no evidence of disease (NED) or with minimal residual disease; patients with stage I uterine serous carcinomas or sarcomas are also eligible after completion of standard therapy * Breast cancer: patients can enter study after completion of all chemotherapy (including trastuzumab), radiation, and breast/axillary surgery; patients may participate while on endocrine therapy; stages I-III patients with the following characteristics: * Estrogen-receptor (ER) negative with positive lymph nodes; ER negative with negative nodes if tumor \> 2 cm; ER positive with positive lymph nodes; and ER positive with negative lymph nodes and tumor \> 5 cm * Sarcomas: patients with sarcomas of any site, who have completed standard therapy, and are in remission, or have minimal disease burden * Lungs: resected patients with hilar or ipsilateral mediastinal nodal disease (i.e., a subset of patients with stage II and IIIA disease); and patients with residual disease on imaging after definitive radiation or chemoradiation therapy * Esophageal: resected patients with any nodal (i.e., thoracic or abdominal) disease; and patients with residual disease on imaging after definitive chemoradiation therapy * Melanoma: stage IIB, stage IIC, and stage III who have completed planned definitive therapy for their disease including radiotherapy and/or interferon; patients declining interferon or with contra-indications to interferon will also be eligible provided they meet requisite criteria for this study (i.e., non-measurable disease); stage IV melanoma of M1a sub-type only, who are not candidates for additional therapy of curative potential (i.e., small volume disease; may be measurable or evaluable); and stage IV melanoma, NED, status post (s/p) complete resection of known sites of disease (i.e., non-measurable disease) * Hepatocellular carcinoma (HCC): patients who have been treated with surgical resection for HCC; and following chemoembolization as adjuvant therapy for HCC * Gastrointestinal: patients who have completed standard therapies for gastric and colorectal cancers, and deemed to be at high-risk of relapse * Any human leukocyte antigen (HLA) type; historic HLA typing is permitted * Tumor expression of NY-ESO-1 or LAGE-1 by immunohistochemistry (IHC) and/or reverse transcription polymerase chain reaction (RTPCR) * Life expectancy \> 6 months * Absolute neutrophil count (ANC) \>= 1,000/uL * Platelets (PLT) \>= 75,000/uL * Hemoglobin (Hgb) \>= 8 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Serum aspartate aminotransferase (serum glutamic oxaloacetic transaminase \[SGOT\]/aspartate aminotransferase \[AST\]) or serum alanine aminotransferase (serum glutamate pyruvate transaminase \[SGPT\]/alanine aminotransferase \[ALT\]) =\< 3 x ULN * Serum creatinine =\< 2 x ULN * Prothrombin time (PT)/international normalized ratio (INR) =\< 1.5 x ULN; patients receiving anticoagulation therapy, PT/INR =\< 3 * Pulmonary function tests: forced expiratory volume in one second (FEV1) \> 50% and diffusion capacity of the lungs for carbon monoxide (DLCO) \> 50% * Pulse oximetry: oxygen (O2) saturation \>= 90% on room air * Electrocardiogram, showing no clinical significant or acute abnormality * Have been informed of other treatment options * Patient or legal representative must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure * Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Patients of child-bearing potential must agree to use acceptable contraceptive methods (e.g., double barrier) during treatment Exclusion Criteria: * Metastatic disease to the central nervous system for which other therapeutic options, including radiotherapy, may be available * Other serious illnesses (e.g., serious infections requiring antibiotics, bleeding disorders) * History of severe autoimmune disorders requiring use of steroids or other immunosuppressives * Concomitant systemic treatment with corticosteroids, anti-histamine or non-steroidal anti-inflammatory drugs, aspirin \> 325 mg; specific cyclooxygenase (COX)-2 inhibitors are permitted * Chemotherapy, radiation therapy, or immunotherapy within 4 weeks prior to first dosing of study agent (6 weeks for nitrosoureas); concomitant hormonal therapies for breast and prostate cancers are allowed * Clinically significant heart disease (New York Heart Association \[NYHA\] class III or IV) within 6 months * Mental impairment that may compromise the ability to give informed consent and comply with the requirements of the study * Lack of availability of a patient for immunological and clinical follow-up assessment * Known pulmonary hypertension * Known hypersensitivity to sirolimus * Evidence of current drug or alcohol abuse or psychiatric impairment, which in the investigator's opinion will prevent completion of the protocol therapy or follow-up * Pregnant or nursing female patients * Unwilling or unable to follow protocol requirements * Any condition which in the investigator's opinion deems the patient an unsuitable candidate to receive study drug; (i.e., any significant medical illness or abnormal laboratory finding that would, in the investigator's judgment, increase the subject's risk by participating in this study) * Received an investigational agent within 30 days prior to enrollment * Known hepatitis B, hepatitis C, or human immunodeficiency virus (HIV)