A Phase II Study of Tivozanib in Patients With Metastatic and Non-resectable Soft Tissue Sarcomas

Inclusion Criteria: * Patients must have histologically or cytologically confirmed sarcoma of soft tissue * Patients must have metastatic and/or locally advanced or locally recurrent disease * Patients must have measurable disease within 4 weeks prior to registration by RECIST 1.1, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan; tumor lesions that are situated in a previously irradiated area may be considered measurable for the purposes of this study only if there is evidence of growth of the area following a course of irradiation that cannot be attributed to necrosis or bleeding into the tumor * Patients must have had a minimum of 1 and a maximum of 4 prior chemotherapy regimens for recurrent/metastatic disease (it will be up to the treating investigator to define what constitutes a "regimen" in each case); the last dose of systemic therapy (include tyrosine kinase inhibitors) must have been given at least 4 weeks prior to initiation of therapy; patients receiving BCNU or mitomycin C must have received their last dose of such therapy at least 6 weeks prior to initiation of therapy * Patients with brain metastasis that have been treated with definitive surgery or radiation and have been clinically stable for 3 months following the procedure with no neurological signs or symptoms and no requirement for systemic glucocorticoids are eligible for study * Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count \>= 1.5 x 10\^9/l * Platelets \>= 75 x 10\^9/l * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (except for patients with known Gilbert Syndrome) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN * Serum creatinine =\< 1.5 x ULN * If urine protein:creatinine (UPC) \>= 1, then a 24-hour urine protein must be assessed; patients must have a 24-hour urine protein value \< 1g to be eligible * Patients must not have current evidence of another malignancy; there are no restrictions regarding prior history of malignancy * If female and of childbearing potential, documentation of negative pregnancy test is required within 7 days prior to first dose; sexually active males and females of childbearing potential must agree to use adequate contraceptive measures, while on study and for 30 days after the last dose of study drug; all subjects (and their partners) must agree to use a highly effective method of contraception; effective birth control includes (a) intrauterine device (IUD) plus one barrier method; or (b) 2 barrier methods; effective barrier methods are male or female condoms, diaphragms, and spermicides (creams or gels that contain a chemical to kill sperm) * Note: oral, implantable, or injectable contraceptives may be affected by cytochrome P450 interactions, and are not considered effective for this study * Adverse events related to prior tumor-specific therapy must have resolved to less than or equal to National Cancer Institute (NCI) Common Terminology Criteria of Adverse Events (CTCAE) (version 4) grade 1 prior to study entry (except alopecia) * Patients taking cytochrome P450 (CYP)3A4 inducers at the time of screening/registration are still eligible for participation, but whenever possible these medications should be discontinued or changed to one that is not an inducer * NOTE: No washout period is required * NOTE: It will be up to the treating investigator's discretion to assess whether or not the risk of discontinuing or changing the medication is higher or lower than the risk of continuing it while on study for the individual patient * Patients must have the ability to understand and the willingness to sign a written informed consent document; signed and dated informed consent must be obtained prior to registration on trial Exclusion Criteria: * Patients with any one of the following sarcoma histological subtypes will not be eligible for participation: alveolar soft-part sarcoma, chondrosarcoma, dermatofibrosarcoma, Ewing sarcoma, gastrointestinal stromal tumor (GIST), Kaposi sarcoma, mixed mesodermal tumor/carcinosarcoma, osteosarcoma, and low grade (grade 1) sarcomas; NOTE: Myxoid liposarcoma with t(12;16) or t(22;22) is permitted; rhabdomyosarcoma (Embryonal, Alveolar, pleomorphic), interdigitating dendritic sarcoma, giant cell tumor of bone * Patients who have had major surgery within 21 days or those who have not recovered from adverse events associated with surgery to =\< grade 1 will not be eligible for participation; excluded from such considerations are surgical changes not expected to improve, e.g. removal of muscle tissue; patients may be on replacement glucocorticoids for pre-existing glucocorticoid deficiency (e.g. Addison's disease) * Patients receiving any other investigational agents will not be eligible for participation * Patients with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to tivozanib will not be eligible for participation * Patients with serious and/or unstable pre-existing medical, psychiatric, or other conditions that could interfere with safety, provision of informed consent, or compliance to study procedures and requirements will not be eligible for participation, including but not limited to: * Uncontrolled intercurrent illness * Ongoing or active infection including HIV, active hepatitis B or C) * Symptomatic congestive heart failure * Unstable angina pectoris * Cardiac arrhythmia * Psychiatric illness/social situations that would limit compliance with study requirements will not be eligible for participation * Pregnant women and women who are breast-feeding will not be eligible for participation * Patients with a history or clinical evidence of central nervous system (CNS) metastases or leptomeningeal carcinomatosis will not be eligible for participation; NOTE: Individuals who have previously-treated CNS metastases, are asymptomatic, and have had no requirement for steroids or anti-seizure medications for 3 months prior to first dose of study drug are the exception; screening with CNS imaging studies such as CT or magnetic resonance imaging (MRI) is required only if clinically indicated or if the subject has a history of CNS metastases * Patients with clinically significant gastrointestinal (GI) abnormalities that may increase the risk for GI bleeding will not be eligible for participation; these include, but are not limited to: * Active peptic ulcer disease * Known intraluminal metastatic lesion/s with risk of bleeding * Inflammatory bowel disease (e.g. ulcerative colitis, Crohn's disease), or other GI conditions with increased risk of perforation * History of abdominal fistula, GI perforation, or intra-abdominal abscess within 28 days prior to beginning study treatment * Clinically significant (\> 1/2 teaspoon) hemoptysis or gastrointestinal hemorrhage in the past 6 months * Patients with evidence of active bleeding or bleeding diathesis will not be eligible for participation; recent hemoptysis would be \>= 1/2 teaspoon of red blood within 8 weeks before first dose of study drug * Patients with clinically significant GI abnormalities that may affect absorption of investigational product will not be eligible for participation; these include but are not limited to: * Malabsorption syndrome * Major resection of the stomach or small bowel * Patients with a corrected QT interval (QTc) \> 480 msecs using Bazett's formula (QT Interval / square root(RR interval)) will not be eligible for participation * Patients with left ventricular ejection fraction (LVEF) \< 50% will not be eligible for participation * NOTE: patients who do not meet the cutoff for LVEF may be re-screened at a later date and, if eligible then, may be enrolled in the study * Patients with a history of any one or more of the following cardiovascular conditions within the past 6 months will not be eligible for participation: * Cardiac angioplasty or stenting * Myocardial infarction * Unstable angina * Coronary artery bypass graft surgery * Symptomatic peripheral vascular disease\\ * Class III or IV congestive heart failure, as defined by the New York Heart Association (NYHA) * Patients with poorly controlled hypertension \[defined as systolic blood pressure (SBP) of \>= 140 mmHg or diastolic blood pressure (DBP) of \>= 90mmHg\] will not be eligible for participation * Note: Initiation or adjustment of antihypertensive medication(s) is permitted prior to study entry; following antihypertensive medication initiation or adjustment, blood pressure (BP) must be re-assessed three times at approximately 2-minute intervals; at least 24 hours must have elapsed between anti-hypertensive medication initiation or adjustment and BP measurement; these three values will be averaged to obtain the mean diastolic blood pressure and the mean systolic blood pressure; the mean SBP / DBP ratio must be \< 140/90 mmHg in order for a subject to be eligible for the study * Patients with cerebrovascular accident including transient ischemic attack (TIA), pulmonary embolism or untreated deep venous thrombosis (DVT) within the past 6 months will not be eligible for participation * Note: subjects with recent DVT who have been therapeutically coagulated for at least 6 weeks are eligible * Patients who had major surgery or trauma within 28 days prior to first dose of investigational product and/or presence of any non-healing wound, fracture, or ulcer (procedures such as catheter placement not considered to be major surgery) will not be eligible for participation * Patients with known endobronchial lesions and/or lesions infiltrating major pulmonary vessels will not be eligible for participation * Note: tumor abutting the vessel is acceptable, but contiguous tumor and vessel is not; CT with contrast is strongly recommended to evaluate such lesions