Evaluating Efficacy of Tivozanib (AV-951) in Biliary Tract Cancers

* INCLUSION CRITERIA: 1. Participants with histologically or cytologically confirmed biliary tract cancer (BTC) (cholangiocarcinoma or gallbladder cancer). Archival tumor sample may be used but if archival tissue is not available or is not adequate, tissue biopsy will be required. 2. Participants must have disease that is not amenable to resection. 3. Participants must have had prior treatment with 1st line chemotherapy. 4. Disease must be measurable by Response Evaluation Criteria in Solid Tumors (RECIST) criteria Version 1.1. 5. Age \>=18 years. NOTE: Because no dosing or adverse event data are currently available on the use of tivozanib in participants \< 18 years of age, children are excluded from this study, but may be eligible for future pediatric trials. 6. ECOG performance status \<= 2 7. Adequate organ and marrow function as defined below: * Hemoglobin \>= 8.0 g/dL * Absolute Neutrophil Count \>= 1,000/mcL * Platelets \>= 75,000/mcL * Total Bilirubin \<= 2.5 X institutional upper limit of normal (ULN) * AST(SGOT)/ALT(SGPT) \<= 5 X institutional ULN * Creatinine Clearance \> 30 * Serum Albumin (g/L) \> 28 8. Negative serum or urine pregnancy test at screening for individuals of childbearing potential (IOCBP), excepting identified false-positive pregnancy test results as permitted in the note below. NOTE: IOCBP is defined as any individual who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. IOCBP must have a negative pregnancy test (HCG blood or urine) during screening. NOTE: Advanced biliary tract disease may secrete hormones that produce false-positive pregnancy test results. A false-positive result will be explicitly determined in this protocol at screening via a series of serial blood tests (i.e., serum HCG measurements) over a 5-day period (i.e., a minimum of a blood test on the first and fifth day of the 5-day period), in which a false-positive result not compatible with pregnancy will be defined as results indicating a consecutive, clinically low, constant level (i.e., no more than a 15% rate of increase) of HCG over the testing period. An ultrasound may be performed for clarification purposes as necessary. 9. All participants (regardless of childbearing potential) must (all) agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 1 month after completion of treatment. 10. Ability of participant to understand and the willingness to sign a written informed consent document. 11. Ability and willingness to co-enroll on the tissue collection protocol 13C0176, "Tumor, Normal Tissue and Specimens from Patients Undergoing Evaluation or Surgical Resection of Solid Tumors". EXCLUSION CRITERIA: 1. Chemotherapy, small molecule or radiation therapy within 2 weeks prior to administration of first dose of study drug. 2. Prior treatment with Tivozanib. 3. History of hepatic encephalopathy within past 12 months or requirement for medications to prevent or control encephalopathy (e.g., no lactulose, rifaximin, etc. if used for purposes of hepatic encephalopathy). 4. Inadequate recovery from any prior surgical procedure or major surgical procedure within 4 weeks prior to administration of first dose of study drug. 5. Previous malignant disease other than the target malignancy within the last 3 years with the exception of basal or squamous cell carcinoma of the skin, cervical carcinoma in situ, chronic lymphocytic leukemia, or thyroid carcinoma. 6. Current active second primary malignancy, other than skin carcinoma (basal or squamous cell carcinoma), chronic lymphocytic leukemia not requiring active treatment, or differentiated thyroid carcinoma. 7. History of allergic reactions or known or suspected hypersensitivity attributed to compounds of similar chemical or biologic composition to tivozanib. 8. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring systemic therapy (see exceptions below), or psychiatric illness/social situations that would limit compliance with study requirements * Human immunodeficiency virus (HIV)-infected participants on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. * For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable and on suppressive therapy, if indicated. For participants with HBV infection who are currently on treatment, they are eligible if they have an undetectable HBV viral load. * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. 9. Significant cardiovascular disease, including: Active clinically symptomatic left ventricular failure, uncontrolled hypertension, myocardial infarction, severe angina, or unstable angina within 3 months prior to administration of first dose of study drug, history of serious ventricular arrhythmia, cardiac arrhythmias requiring anti-arrhythmic medications. 10. Uncontrolled hypertension, i.e., blood pressure (BP) of \>= 150/90 mmHg; participants who have a history of hypertension controlled by medication must be on a stable dose of antihypertensive therapy such that there has been no increase in hypertensive medications or dosage (for at least -14 days) and meet all other inclusion criteria. 11. Significant hematologic, gastrointestinal, thromboembolic, vascular, bleeding, or coagulation disorders. 12. GI Bleeding (e.g., esophageal varices or ulcer bleeding) within 3 months. (Note: For participants with a history of GI bleeding for more than 12 months or assessed as high risk for esophageal variceal by the Investigator, adequate endoscopic therapy according to institutional standards is required.) 13. Complex biliary obstruction requiring bile duct stents at more than one level of the biliary tree or external biliary drainage. 14. Recurrent episodes of cholangitis (\>1) in the preceding 3 months prior to enrollment. 15. Therapeutic anti-coagulation or anti-platelet therapy with the exception of low molecular weight heparin, aspirin, or factor Xa inhibitors. 16. Pregnant or lactating individuals. Pregnant individuals are excluded from this study because based on findings in animals and its mechanism of action, tivozanib can cause fetal harm when administered to a pregnant individual. In animal reproduction studies, administration of tivozanib to pregnant rats caused adverse developmental outcomes including embryo- fetal mortality. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the individual with tivozanib, nursing (such as breastfeeding) should be discontinued if the individual is treated with tivozanib. These potential risks may also apply to other agents used in this study.