Cabozantinib S-malate in Treating Patients With Relapsed Osteosarcoma or Ewing Sarcoma

Inclusion Criteria: * Young patient age between 12 - 15 could be included in only 6 centers (Bordeaux, Lyon, Villejuif, Lille, Marseille and Paris) * Patients must have histologically confirmed diagnosis of osteosarcoma or Ewing sarcoma by central review, except if the diagnosis was already confirmed by the RRePS (Reseau de Reference en Pathologie des Sarcomes et des Tissus Mous et des Visceres) network * Relapsed disease after standard chemotherapy * Patients must have measurable disease (lesion in previously irradiated field could be considered as measurable if progressive at inclusion) defined as per RECIST v1.1 with at least one lesion that can be measured in at least one dimension (longest diameter to be recorded) as \>= 10 mm with spiral computed tomography (CT) scan * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 * Life expectancy of greater than 3 months * Absolute neutrophil count \>= 1,500/mcL * Lymphocyte count \> 1,000/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional upper limit of normal * Creatinine =\< 1.5 x ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal (Cockcroft formula) * Hemoglobin \>= 9 g/dL * Serum albumin \>= 2.8 g/dL * Lipase \< 2.0 x ULN and no radiologic or clinical evidence of pancreatitis * Urine protein/creatinine ratio (UPCR) =\< 1 * Serum phosphorus \>= lower limit of normal (LLN) * Serum calcium \>= LLN * Serum magnesium \>= LLN * Serum potassium \>= LLN * Female subjects of childbearing potential must not be pregnant at screening; females of childbearing potential are defined as premenopausal females capable of becoming pregnant (ie, females who have had any evidence of menses in the past 12 months, with the exception of those who had prior hysterectomy); however, women who have been amenorrheic for 12 or more months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, antiestrogens, low body weight, ovarian suppression or other reasons * The effects of Cabozantinib on the developing human fetus are unknown; for this reason and because tyrosine kinase inhibitors agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (see below) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of cabozantinib administration * Sexually active subjects (men and women) must agree to use medically accepted barrier methods of contraception (e.g., male or female condom) during the course of the study and for 4 months after the last dose of study drug(s), even if oral contraceptives are also used; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control during the course of the study and for 4 months after the last dose of study drug(s) * Metastatic or unresectable locally advanced * Documented disease progression (as per RECIST v1.1) before study entry; for patients with osteosarcoma, this progression will be confirmed by central review on the basis of two CT scan or magnetic resonance imaging (MRI) obtained at less than 6 months in the period of 12 months prior to inclusion * Ability to understand and the willingness to sign a written informed consent document * In accordance with French Regulatory Authorities: Patients with French Social Security in compliance with the French law relating to biomedical research (Huriet Law 88-1138 and related decrees) Exclusion Criteria: * The subject has received cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (e.g., cytokines or antibodies) within 3 weeks, or nitrosoureas/mitomycin C within 6 weeks before the first dose of study treatment * Prior treatment with cabozantinib * Radiation therapy for bone metastasis within 2 weeks, any other external radiation therapy within 4 weeks before the first dose of study treatment; systemic treatment with radionuclides within 6 weeks before the first dose of study treatment; subjects with clinically relevant ongoing complications from prior radiation therapy are not eligible * Receipt of any type of small molecule kinase inhibitor (including investigational kinase inhibitor) within 14 days before the first dose of study treatment; note: subjects with prostate cancer currently receiving luteinizing hormone-releasing hormone (LHRH) or gonadotropin-releasing hormone (GnRH) agonists may be maintained on these agents * The subject has received any other type of investigational agent within 28 days before the first dose of study treatment * The subject has not recovered to baseline or Common Terminology Criteria for Adverse Events (CTCAE) =\< grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant adverse events (AEs) * The subject has a primary brain tumor * Known brain metastases or cranial epidural disease unless adequately treated with radiotherapy and/or surgery (including radiosurgery) and stable for at least 2 weeks before the first dose of study treatment; eligible subjects must be neurologically asymptomatic and without corticosteroid treatment at the time of the start of study treatment * The subject has prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin time (PTT) test \>= 1.3 x the laboratory ULN within 7 days before the first dose of study treatment * The subject requires concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, thrombin or factor Xa inhibitors, or antiplatelet agents (e.g., clopidogrel); low dose aspirin (=\< 81 mg/day), low-dose warfarin (=\< 1 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * The subject requires chronic concomitant treatment of strong cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital, and St. John's wort) * it is important to regularly consult a frequently-updated list; medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * The subject has experienced any of the following: * Clinically-significant gastrointestinal bleeding within 6 months before the first dose of study treatment * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months before the first dose of study treatment * Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment * The subject has radiographic evidence of cavitating pulmonary lesion(s) * The subject has tumor in contact with, invading or encasing any major blood vessels * The subject has evidence of tumor invading the gastrointestinal (GI) tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before the first dose of cabozantinib * The subject has uncontrolled, significant intercurrent or recent illness including, but not limited to, the following conditions: * Cardiovascular disorders including: * Congestive heart failure (CHF): New York Heart Association (NYHA) class III (moderate) or class IV (severe) at the time of screening * Concurrent uncontrolled hypertension defined as sustained blood pressure (BP) \> 150 mm Hg systolic, or \> 90 mm Hg diastolic despite optimal antihypertensive treatment within 7 days of the first dose of study treatment * Any history of congenital long QT syndrome * Any of the following within 6 months before the first dose of study treatment: * Unstable angina pectoris * Clinically-significant cardiac arrhythmias * Stroke (including transient ischemic attack \[TIA\], or other ischemic event) * Myocardial infarction * Thromboembolic event requiring therapeutic anticoagulation (note: subjects with a venous filter \[e.g. vena cava filter\] are not eligible for this study) * Gastrointestinal disorders particularly those associated with a high risk of perforation or fistula formation including: * Any of the following within 28 days before the first dose of study treatment * Intra-abdominal tumor/metastases invading GI mucosa * Any evidence of active peptic ulcer disease, patients must be completely recovered * Any evidence of inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, patients must be completely recovered from these conditions * Malabsorption syndrome * Any of the following within 6 months before the first dose of study treatment: * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * Intra-abdominal abscess; note: complete resolution of an intra-abdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more than 6 months before the first dose of study treatment * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months before the first dose of study therapy * Other clinically significant disorders such as: * Active infection requiring systemic treatment within 28 days before the first dose of study treatment * Serious non-healing wound/ulcer/bone fracture within 28 days before the first dose of study treatment * History of organ transplant * Concurrent uncompensated hypothyroidism or thyroid dysfunction within 7 days before the first dose of study treatment * History of major surgery as follows: * Major surgery within 12 weeks before the first dose of study treatment; complete wound healing from major surgery must have occurred 1 month before the first dose of study treatment * Minor surgery (including uncomplicated tooth extractions) within 28 days before the first dose of study treatment with complete wound healing at least 10 days before the first dose of study treatment; subjects with clinically relevant ongoing complications from prior surgery are not eligible * The subject is unable to swallow tablets * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before treatment; note: if initial QTcF is found to be \> 500 ms, two additional electrocardiograms (EKGs) separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard * The subject is unable or unwilling to abide by the study protocol or cooperate fully with the investigator or designee * The subject has had evidence within 2 years of the start of study treatment of another malignancy which required systemic treatment * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cabozantinib * Pregnant women are excluded from this study because cabozantinib is a tyrosine kinase inhibitor with 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 cabozantinib, breastfeeding should be discontinued if the mother is treated with cabozantinib; these potential risks may also apply to other agents used in this study * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with cabozantinib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated * Participation to a study involving a medical or therapeutic intervention in the last 30 days * Prior participation in this study