Nab-Sirolimus and Pazopanib Hydrochloride in Treating Patients With Advanced Nonadipocytic Soft Tissue Sarcomas

Inclusion Criteria: * Subjects, \>= 18 years old, must have a histologically confirmed diagnosis of non-adipocytic soft tissue sarcoma (STS) that is either metastatic or locally advanced and for which curative therapy is not available, surgery is not a recommended option, and pazopanib treatment is indicated. * Subjects must have one or more measurable target lesions by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1, assessed via computed tomography (CT) scan or magnetic resonance imaging (MRI). * Clinical or radiological progression or failure due to toxicity on at least 1 prior regimen of systemic treatment for advanced disease. Subjects may not have received more than 4 prior lines of systemic therapy (no more than 2 prior therapies may be combination cytotoxic therapies). Neo-adjuvant/adjuvant/maintenance treatments are not included for this criterion. * Last dose of prior therapy must have been completed a minimum of 14 days prior to start of protocol therapy. All ongoing toxicities related to prior therapy must be resolved or grade 1 (except alopecia). \* NOTE: Toxicities from prior therapy that have resolved with sequelae (e.g. hypothyroidism) and are asymptomatic or well-controlled are not exclusionary. * Total bilirubin =\< upper limit of normal (ULN) mg/dL (Subjects with known Gilbert's syndrome and a total bilirubin =\< 3 mg/dl are permitted to enroll to phase 2/expansion phase only with sponsor-investigator approval). * Aspartate aminotransferase (AST) =\< 2.5 x ULN and alanine aminotransferase (ALT) =\< 2.5 x ULN. * Serum creatinine =\<1.5 x ULN (If serum creatinine is \> 1.5 mg/dL, calculated creatinine clearance \> 50 mL/min using the Cockcroft-Gault formula may be included). * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L. * Platelet count \>= 100,000/mm\^3 (100 x 10\^9/L). * Hemoglobin \>= 9 g/dL. * Serum triglyceride =\< 300 mg/dL. * Serum cholesterol =\< 350 mg/dL. * Baseline cardiac left ventricular ejection fraction (LVEF) within institutional limits of normal (by echocardiogram or multigated acquisition \[MUGA\] study). * Baseline electrocardiogram with corrected QT (QTc) \< 480 millisecond (Bazett's). * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1. * Male or non-pregnant and non-breast feeding female: * Females of child-bearing potential must agree to use highly effective contraception without interruption from initiation of therapy and while on study medication and have a negative serum pregnancy test (beta human chorionic gonadotropin \[beta-hCG\]) result at screening and agree to ongoing pregnancy testing during the course of the study, and at the end of study treatment. A highly effective method of contraception is defined as one that results in a low failure rate (that is, \< 1% per year), when used consistently and correctly, such as implants, injectables, combined oral contraceptives, some intrauterine contraceptive devices, sexual abstinence, or a vasectomized partner. * Male patients must practice abstinence or agree to use a condom during sexual contact with a pregnant female or a female of childbearing potential while participating in the study. * Life expectancy of \> 3 months, as determined by the investigator. * Ability to understand and sign informed consent. * Willingness and ability to comply with scheduled visits, laboratory tests, and other study procedures. Exclusion Criteria: * Soft tissue sarcomas with biology or defined treatments for which pazopanib is not indicated, including adipocytic STS, gastrointestinal stromal tumors (GIST), or Kaposi's sarcoma. * Previously received an mTOR (mammalian target of rapamycin) inhibitor or angiogenesis inhibitor. * Known active uncontrolled or symptomatic central nervous system (CNS) metastases. A subject with controlled and asymptomatic CNS metastases may participate in this study. As such, the patient must have completed any prior treatment for CNS metastases \>= 28 days (including radiotherapy and/or surgery) prior to start of treatment in this study and should not be receiving chronic corticosteroid therapy for the CNS metastases. * Subjects with hemoptysis, central nervous system hemorrhage or gastrointestinal hemorrhage within the last 6 months prior to treatment are excluded due to pazopanib-associated risk of bleeding. * Subjects with severe hepatic impairment and active gastrointestinal bleeding. * Uncontrolled serious medical or psychiatric illness. * Subjects with a currently active second malignancy other than non-melanoma skin cancers, carcinoma in situ of the cervix, resected incidental prostate cancer, or other adequately treated carcinoma-in-situ are ineligible. Subjects are not considered to have a currently active malignancy if they have completed therapy and are free of disease for \>= 1 year). * Recent infection requiring systemic anti-infective treatment that was completed =\< 14 days prior to enrollment (with the exception of uncomplicated urinary tract infection or upper respiratory tract infection). * No clinically significant gastrointestinal abnormalities including malabsorption syndrome, major resection of the stomach or small bowel that could affect the absorption of study drug, active peptic ulcer disease, inflammatory bowel disease, ulcerative colitis, or other gastrointestinal conditions with increased risk of perforation, history of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days prior to beginning study treatment. * Uncontrolled diabetes mellitus as defined by hemoglobin A1C (HbA1c) \> 8% despite adequate therapy. * Subjects with unstable coronary artery disease, myocardial infarction, or an arterial thromboembolic event during preceding 6 months. * Subjects with history of interstitial lung disease and/or pneumonitis, or pulmonary hypertension. * Use of strong inhibitors and inducers of CYP3A4 within the 14 days prior to receiving the first dose of nab-sirolimus. Additionally, use of any known CYP3A4 substrates with narrow therapeutic window (such as fentanyl, alfentanil, astemizole, cisapride, dihydroergotamine, pimozide, quinidine, terfenadine) within the 14 days prior to receiving the first dose of nab-sirolimus. * Active hepatitis B or hepatitis C infection. * Systemic immunosuppression, including human immunodeficiency virus (HIV) positive status with or without acquired immunodeficiency syndrome (AIDS). * Subjects with history of intestinal perforations, fistula, hemorrhages and/or hemoptysis =\< 6 months prior to first study treatment. * Subjects with hypercholesterolemia receiving ongoing treatment with simvastatin. * Subjects who have had major surgery within 28 days of planned initiation of protocol therapy, or patients who have/have had wound dehiscence, or other open wounds (including diabetic or infectious wounds) with active wound complications. * Subjects with prior history of severe hypersensitivity (grade 3 or higher) to any known drug excipients, including anaphylaxis to human serum albumin. * Subjects with uncontrolled hypertension, defined as an average systolic blood pressure (SBP) \>= 140 mmHg or an average diastolic blood pressure (DBP) \>= 90 mmHg despite best supportive care measures.