Testing the Addition of an Anti-cancer Drug, Sapanisertib, to the Usual Chemotherapy Treatment (Cabozantinib) in Metastatic Liver Cell Cancer With a Change in Genes for the Protein β-Catenin, The SAPHIRE Trial

Inclusion Criteria: * Patients must have histologically or cytologically confirmed HCC, not amenable to curative treatment approach * For Phase 2, patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray or as ≥ 10 mm (≥ 1 cm) with CT scan, MRI, or calipers by clinical exam * For phase 2, patients must have a β-catenin mutation, based on next generation eequencing (NGS) testing through Clinical Laboratory Improvement Amendments (CLIA)-certified commercially available standard of care assay * Patients must have received at least one prior line of systemic therapy in the metastatic setting, including a prior immune checkpoint inhibitor therapy unless not eligible. For the phase 2 portion, patients must have received at least one and no more than two prior lines of systemic therapy in the metastatic setting, including a prior immune checkpoint inhibitor therapy unless not eligible * Age ≥ 18 years. Because no dosing or adverse event data are currently available on the use of sapanisertib in combination with cabozantinib in patients \<18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 (Karnofsky ≥ 50%) * Child Pugh score of A * Absolute neutrophil count ≥ 1,000/mcL * Platelets ≥ 30,000/mcL * Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) ≤ 5 × institutional ULN * Glomerular filtration rate (eGFR) ≥ 40 mL/min/1.73 m\^2 * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better * For the phase 2 portion, availability of archival tumor tissue at the time of patient enrollment for banking for molecular profiling studies * The effects of sapanisertib and cabozantinib on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and after completion of drug administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Both men and women treated or enrolled on this protocol must agree to use adequate contraception prior to the study, for the duration of study participation, and for the following duration after completion of sapanisertib and cabozantinib administration: * 90 days and 120 days after last dose of sapanisertib for women of childbearing potential and men respectively, * 5 months and 7 months after last dose of cabozantinib for women of childbearing potential and men respectively * Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants Exclusion Criteria: * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to sapanisertib and cabozantinib * Use of strong CYP3A4-inhibiting agents due to drug-drug interaction with cabozantinib * Prior exposure to cabozantinib * Patients who are unable to swallow oral medications such as capsules and tablets and patients with gastrointestinal conditions that may affect the absorption of oral medications * Patients with uncontrolled intercurrent illness or any other significant condition(s) that would make participation in this protocol unreasonably hazardous * Pregnant women are excluded from this study because sapanisertib and cabozantinib have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with sapanisertib and cabozantinib, breastfeeding should be discontinued if the mother is treated with sapanisertib and cabozantinib