Nanoparticle Albumin-Bound Rapamycin in Treating Patients With Advanced Cancer With mTOR Mutations

Inclusion Criteria: * Histologic proof of cancer which is now not amenable to curative standard treatment options * Patient must have received at least 1 prior standard therapy for their disease * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Ability to provide informed written consent * Willing to return to enrolling institution for follow-up (active monitoring phase of the study); Note: During the active monitoring phase of a study (i.e., active treatment), participants must be willing to return to the consenting institution for follow-up * Life expectancy \>= 84 days (3 months) * Identification of a drug target/targets through molecular profiling performed as a part of routine clinical care and treatment recommendation by the Mayo Clinic Genomics Tumor Board (GTB); NOTE: If profile matches more than 1 treatment arm, final decision for treatment arm assignment to be made by patients treating physician; it will be required for the genomic aberration to be identified through a test in a Clinical Laboratory Improvement Amendments (CLIA) workflow; assays used will range from single gene abnormalities (e.g. fluorescent in situ hybridization \[FISH\] for human epidermal growth factor receptor 2 \[ERBB2\] amplifications) to next generation sequencing based gene panels (Foundation One®) to more comprehensive assays such as whole exome sequencing; the Mayo Clinic GTB will serve as the centralized point of data synthesis to allow for assessment of molecular profiling accomplished through a heterogeneous array of tests * Date of Mayo Clinic Genomics Tumor Board review =\< 3 months prior to registration * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 criteria (for solid tumors) or equivalent criteria (for patients with non-solid tumor malignancies) * Patient meets all sub-protocol specific criteria of each applicable sub-protocol * Ability to complete questionnaire(s) by themselves or with assistance * SUB-PROTOCOL AIM A: Histological confirmation of renal cell carcinoma, head and neck cancer, endometrial cancer, breast cancer, ovarian cancer, prostate cancer, squamous cell cervical or uterine cancer, or bladder cancer * SUB-PROTOCOL AIM A: Confirmation of advanced cancer with mTOR pathway aberrations as determined through routine clinical care using pathway aberrations performed in a CLIA certified laboratory; cancer genomic profiling tests incorporating next generation sequencing from archival formalin-fixed paraffin-embedded tissue (FFPE) are validated with sensitivities and specificities of 99% and 99%, respectively; in the assay, hybrid-capture-selected deoxyribonucleic acid (DNA) libraries are sequenced to depths targeting \> 500 × coverage by non-polymerase chain reaction (PCR) duplicate read pairs, with \> 99% of exons at coverage \> 100 ×); multiplatform profiling may include immunohistochemistry and in situ hybridization methods with previously established negative/positive cutoffs performed in a CLIA certified lab; at least one pathway aberration must be identified; these must be confirmed in a CLIA certified lab; the potential mTOR aberrations that could be identified are listed below, please note that this list is not all inclusive; if a CLIA validated report lists an mTOR pathway inhibitor as a target drug for a genetic aberration, then it can be considered eligible for the purposes of this study; v-akt murine thymoma viral oncogene homolog 1 (AKT1), MTOR, phosphatidylinositol-4,5-bisphosphate 3-kinase, catalytic subunit alpha (PIK3CA), tuberous sclerosis (TSC)1, TSC2, retrovirus-associated DNA sequence (Ras) homolog enriched in brain (RHEB), serine/threonine kinase 11 (STK11), neurofibromin (NF)1/2 * SUB-PROTOCOL AIM A: Absolute neutrophil count (ANC) \>= 1500/mm\^3 * SUB-PROTOCOL AIM A: Platelet count \>= 100,000/mm\^3 * SUB-PROTOCOL AIM A: Hemoglobin \>= 9.0 g/dL * SUB-PROTOCOL AIM A: Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * SUB-PROTOCOL AIM A: Aspartate transaminase (AST); alanine aminotransferase (ALT) =\< 1.5 x ULN; NOTE: if subject has tumor involvement in the liver =\< 5 X ULN * SUB-PROTOCOL AIM A: Serum cholesterol =\< 350 mg/dL * SUB-PROTOCOL AIM A: Serum triglyceride =\< 300 mg/dL * SUB-PROTOCOL AIM A: Serum creatinine =\< 1.5 x ULN * SUB-PROTOCOL AIM A: Previously treated patients who have failed, unable to tolerate, or refused other available active therapies * SUB-PROTOCOL AIM A: Women of child-bearing potential, defined as sexually mature women who have not undergone a hysterectomy or who have not been naturally postmenopausal for at least 12 consecutive months (e.g., who has had menses any time in the preceding 12 consecutive months), must have a negative serum pregnancy test =\< 14 days prior to registration and must use two forms of highly effective contraception (also applicable to their partners who are biologically able to conceive) * SUB-PROTOCOL AIM A: Adequate coagulation function as defined by either of the following criteria: * International normalized ratio (INR) =\< 1.5 x ULN * For subjects receiving warfarin or low molecular weight heparin (LMWH), the subjects must, in the investigator's opinion, be clinically stable with no evidence of active bleeding while receiving anticoagulant therapy; the INR for these patients may exceed 1.5 x ULN if that is the goal of the anticoagulant therapy Exclusion Criteria: * Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Failure to fully recover from acute, reversible effects of prior chemotherapy (other anti-neoplastic therapy) and radiation therapy to adverse event severity of =\< grade 1 * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * SUB-PROTOCOL AIM A: Any of the following: * Pregnant women * Nursing women * Women of child-bearing potential, who are biologically able to conceive, or men who are able to father a child, not employing two forms of highly effective contraception * Highly effective contraception (e.g., male condom with spermicide, diaphragm with spermicide, intra-uterine device, and total abstinence) must be used by both sexes during the study and must be continued for 6 months after the end of study treatment; Note: Oral, implantable, or injectable hormone contraceptives are not considered effective for this study * SUB-PROTOCOL AIM A: Any of the following treatments: * Chemotherapy within 4 weeks before treatment with nab-rapamycin * Hormonal therapy within 4 weeks before treatment with nab-rapamycin (with the exception of leuprolide, degarelix, or goserelin) * Immunotherapy within 4 weeks before treatment with nab-rapamycin * Radiotherapy within 4 weeks before treatment with nab-rapamycin * Treatment with nitrosoureas, mitomycin, or extensive radiotherapy within 6 weeks before treatment with nab-rapamycin * Immunosuppressive agents within 3 weeks before treatment with nab-rapamycin (except corticosteroids used as antiemetics) * Use of prior mTOR pathway inhibitor therapy * SUB-PROTOCOL AIM A: Patients with a history of interstitial lung disease and/or pneumonia * SUB-PROTOCOL AIM A: Receiving any concomitant antitumor therapy or inhibitors of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) * SUB-PROTOCOL AIM A: History of allergic reactions attributed to compounds of similar chemical or biologic composition including macrolide (e.g. azithromycin, clarithromycin, dirithromycin, and erythromycin) and ketolide antibiotics * SUB-PROTOCOL AIM A: Major surgery (e.g., intra-thoracic, intra-abdominal or intra-pelvic) =\< 4 weeks prior to registration or failure to recover from side effects of such surgery; exceptions: port placements, nephrectomy, tumor biopsies, and minor surgeries * SUB-PROTOCOL AIM A: Concurrent use of any other approved or investigational anticancer agents which would be considered as a treatment for the primary neoplasm * SUB-PROTOCOL AIM A: Patients with a history of alcoholism, drug addiction or psychotic disorders * SUB-PROTOCOL AIM A: Uncontrolled diabetes mellitus as defined by hemoglobin A1C (HbA1c) \> 8% despite adequate therapy; unstable coronary artery disease or myocardial infarction during preceding 6 months; or hypertension uncontrolled by medication * SUB-PROTOCOL AIM A: Patients who required therapeutic doses of anticoagulants