Sapanisertib or Pazopanib Hydrochloride in Treating Patients With Locally Advanced or Metastatic Sarcoma

Inclusion Criteria:

* Patients must have slides available for submission to central pathology review; this review is mandatory prior to registration to confirm eligibility and proper cohort assignment

  * HISTOLOGIC COHORT 1: Undifferentiated pleomorphic sarcoma (includes: malignant fibrous histiocytoma, myxofibrosarcoma, high grade sarcoma not otherwise specified \[NOS\])
  * HISTOLOGIC COHORT 2: Leiomyosarcoma (either uterine or extra-uterine)
  * HISTOLOGIC COHORT 3: Other (either malignant peripheral nerve sheath tumor or synovial sarcoma); during the phase II portion of the study, enrollment will be limited to maximum of 25 patients in this cohort

    * Note that the phase I is limited to the histologic subtypes listed above; since patients will be enrolling onto dose cohorts during the phase I, they will not enroll onto specific histologic cohorts, although the histologic subtype informed will be collected during patient enrollment
* Histologic documentation: Eligible patients must have histopathologically confirmed sarcoma of one of the subtypes listed, by central review
* Locally advanced or metastatic disease; locally advanced disease is defined as disease not amenable to local therapy such as surgery and/or radiation
* Measurable disease
* Progression on at least one prior systemic chemotherapy for advanced, unresectable or metastatic disease; prior adjuvant or neoadjuvant therapy is not included as prior systemic chemotherapy unless treatment occurred within the 6 months prior to study enrollment
* There is no limit to the number of prior lines of treatment a patient has received
* No treatment with biological therapy, immunotherapy, chemotherapy, investigational agent for malignancy, or radiation =\< 28 days before study registration; no treatment with nitrosourea or mitomycin =\< 42 days before study registration
* No treatment with radiation therapy =\< 28 days before study registration
* Patients should have resolution of any toxic effects of prior therapy (except alopecia) to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, grade 1 or less
* Prior treatment with pazopanib or any phosphoinositide 3-kinase (PI3K), mTOR, protein kinase B (AKT), or dual PI3K/mTOR complex (CREB regulated transcription coactivator \[TORC\]1/TORC2) inhibitors will be prohibited
* Not pregnant and not nursing; because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown and an agent that has known genotoxic, mutagenic and teratogenic effects; therefore, for women of childbearing potential only, a negative pregnancy test done =\< 7 days prior to registration is required
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 1
* Patient history: patients who have any of the following are NOT eligible:

  * Central nervous system (CNS): Symptomatic, untreated, or uncontrolled brain metastases present
  * Heme: Active bleeding or bleeding diathesis
  * Gastrointestinal (GI):

    * Abdominal fistula, GI perforation, or intra-abdominal abscess within 28 days prior to registration
    * Acute GI bleed within 28 days of registration
  * Diabetes mellitus: Patients with diabetes mellitus with inadequate control, based on either a glycosylated hemoglobin (Hgb A1c) of \> 7.0 or fasting blood glucose above or equal to 130 mg/dL
  * Cardiac and vascular disorders:

    * History of congenital long QT syndrome or torsades de pointes
    * Any arrhythmia that is currently not rate-controlled (rate between 60 and 100)
    * Prolongation of corrected QT interval via Fridericia's formula (QTcF) \> 480 msec
    * Ongoing unstable angina
    * Symptomatic peripheral vascular disease
    * Arterial thrombosis within 28 days of registration including transient ischemic attack (TIA), cerebrovascular accident (CVA), myocardial infarction (MI)
    * Patients with deep vein thrombosis (DVT) or pulmonary embolism (PE) must be on a stable dose of anticoagulation for 14 days prior to registration
    * Uncontrolled hypertension, defined as blood pressure (BP) \> 140/90
    * Multi gated acquisition scan (MUGA) with ejection fraction (EF), 50% or echocardiogram (echo) with low EF
    * Class III or IV congestive heart failure (CHF) within 28 days of registration
* Chronic concomitant treatment with proton pump inhibitors must discontinue the drug for 7 days prior to registration on the study
* Chronic concomitant treatment with strong inhibitors of CYP3A4 inhibitors must discontinue the drug for 14 days prior to registration on the study
* Chronic concomitant treatment with strong CYP3A4 inducers is not allowed; patients must discontinue the drug 14 days prior to the start of study treatment
* Absolute neutrophil count (ANC) \>= 1,500/mm\^3
* Platelet count \>= 100,000/mm\^3
* Creatinine =\< 1.5 x upper limit of normal (ULN)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN); unless patient has Gilbert disease
* Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3 x upper limit of normal (ULN); if liver metastases, =\< 5 x upper limit of normal (ULN)
* Urine protein creatinine (UPC) =\< 1; if UPC \>= 1, then a 24-hour urine protein must be assessed; eligible patients must have a 24-hour urine protein value \< 1 g/L
* Thyroid stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH however if the Free T4 is normal and patient is clinically euthyroid, patient is eligible