Pazopanib Hydrochloride and Topotecan Hydrochloride in Treating Patients With Metastatic Soft Tissue and Bone Sarcomas

Inclusion Criteria: * Patients must provide written informed consent prior to performance of study-specific procedures or assessments and must be willing to comply with treatment and follow-up * Patients must have a histologically confirmed diagnosis of: * Metastatic soft tissue sarcomas (non-liposarcoma) * Metastatic osteosarcoma * Metastatic liposarcoma- high grade, de-differentiated, or myxoid Note: pathology is not required to be reviewed at the treating institution; a copy of the pathology report is sufficient for eligibility purposes * Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Patients must have measurable disease within 4 weeks prior to registration by Response Evaluation Criteria In Solid Tumors (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 \>= 10mm with spiral computed tomography (CT) scan * 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 investigator to determine what constitutes a "regimen" in each case; the last dose of systemic therapy much have been given at least 4 weeks prior to initiation of therapy; patients receiving BCNU or mitomycin C must have received their last dose at least 6 weeks prior to initiation of therapy * Patients with brain metastasis are eligible for participation only if they have been treated with definitive surgery or radiation (surgery ± radiotherapy, radiosurgery, or gamma knife) and meet both of the following criteria: a) are asymptomatic and b) have no requirement for steroids or enzyme-inducing anticonvulsants in prior 12 week interval * Absolute neutrophil count (ANC) \>= 1.5 X 10\^9/L (tested within 7 days prior to Registration) * Hemoglobin \>= 9 g/dL (5.6 mmol/L) * Subjects may not have had a transfusion within 7 days of screening assessment * Platelets \>= 100 X 10\^9/L * Subjects may not have had a transfusion within 7 days of screening assessment * Prothrombin time (PT) or international normalized ratio (INR) =\< 1.2 X upper limit of normal (ULN) * Subjects receiving anticoagulant therapy are eligible if their INR is stable and within the recommended range for the desired level of anticoagulation * Activated partial thromboplastin time (aPTT) =\< 1.2 X ULN * Total bilirubin =\< 1.5 X ULN * Alanine amino transferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 X ULN * Concomitant elevations in bilirubin and AST/ALT above 1.0 x ULN (upper limit of normal) are not permitted * Serum creatinine =\< 1.5 mg/dL (133 umol/L) or, if \> 1.5 mg/dL: calculated creatinine clearance (ClCR) \>= 30 mL/min to \>= 50 mL/min * Urine protein to creatinine ratio (UPC) \< 1 * If UPC \>= 1, then a 24-hour urine protein must be assessed; subjects must have a 24-hour urine protein value \< 1 g to be eligible; use of urine dipstick for renal function assessment is not acceptable * Females of child-bearing potential (FOCBP) and males must agree to use adequate contraception prior to study entry, for the duration of study participation, and for 30 days following completion of therapy; should a female patient become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; Note: a FOCBP is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) * FOCBP must have a negative pregnancy test within 7 days prior to registration on study * Note: female patients who are lactating should discontinue nursing prior to the first dose of study drug and should refrain from nursing throughout the treatment period and for 14 days following the last dose of study drug * Are able to swallow and retain oral tablets Exclusion Criteria: * Patients with any of the following sarcoma histologic subtypes will not be eligible for participation: * Alveolar soft-part sarcoma * Chondrosarcoma * Dermatofibrosarcoma * Ewing sarcoma * Gastrointestinal stromal tumor (GIST) * Kaposi sarcoma (non-human immunodeficiency virus \[HIV\] and HIV related disease) * Mixed mesodermal tumor/carcinosarcoma * Low grade (grade 1) sarcomas * Rhabdomyosarcoma (embryonal, alveolar, pleomorphic) * Interdigitating dendritic sarcoma * Giant cell tumor of the bone * Patients must not have received prior treatment with pazopanib or topotecan * Patients must not have an active secondary malignancy * Prior malignancy, unless they have been disease-free for 3 years, or have a history of completely resected non-melanomatous skin carcinoma or successfully treated in situ carcinoma * Clinically significant gastrointestinal abnormalities that may increase the risk for gastrointestinal bleeding including, but 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 gastrointestinal conditions with increased risk of perforation * History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days prior to beginning study treatment * Clinically significant gastrointestinal abnormalities that may affect absorption of investigational product including, but not limited to: * Malabsorption syndrome * Major resection of the stomach or small bowel * Corrected QT interval (QTc) \> 480 msecs using Bazett's formula * History of any one or more of the following cardiovascular conditions within the past 12 months: * 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) * Poorly controlled hypertension \[defined as systolic blood pressure (SBP) of \>= 140 mmHg or diastolic blood pressure (DBP) of \>= 90mmHg 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 should 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 (or 150/90 mm Hg, if this criterion deemed safe by principal investigator \[PI\] and the quality assurance monitor \[QAM\]) in order for a patient to be eligible for the study * Patients with a history of cerebrovascular accident including transient ischemic attack (TIA), pulmonary embolism, or untreated deep venous thrombosis (DVT); patients with DVT must have received appropriate therapy for at least 6 months to be considered eligible * 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) * Evidence of active bleeding or bleeding diathesis * Known endobronchial lesions and/or lesions infiltrating major pulmonary vessels that increase the risk of pulmonary hemorrhage * Large protruding endobronchial lesions in the main or lobar bronchi are excluded; however, endobronchial lesions in the segmented bronchi are allowed * Lesions extensively infiltrating the main or lobar bronchi are excluded; however, minor infiltrations in the wall of the bronchi are allowed * Recent hemoptysis (\>= 1/2 teaspoon \[2.5 mL\]) of red blood within 8 weeks before first dose of study drug * Any serious and/or unstable pre-existing medical, psychiatric, or other condition that could interfere with patient's safety, provision of informed consent, or compliance to study procedures * Unable or unwilling to discontinue use of inducers and inhibitors of cytochrome P450 (CYP450) listed for at least 14 days or five half-lives of a drug (whichever is longer) prior to the first dose of study drug and for the duration of the study; cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) substrates can be administered, but investigators will need to be aware of possible increased or decreased effectiveness of the non-study drug and this should be recorded in concomitant medications; breast cancer resistance protein (BCRP) and p-glycoprotein (PgP) inducers and inhibitors will be also prohibited * Treatment with any of the following anti-cancer therapies: * Radiation therapy, surgery or tumor embolization within 14 days prior to the first dose of therapy * Chemotherapy, immunotherapy, biologic therapy, investigational therapy or hormonal therapy within 14 days or five half-lives of a drug (whichever is longer) prior to the first dose of therapy * Administration of any non-oncologic investigational drug within 30 days or 5 half-lives whichever is longer prior to receiving the first dose of study treatment * Any ongoing toxicity related to prior anti-cancer therapy that is \> grade 1 and/or that is progressing in severity (exceptions include alopecia, fatigue, and hematologic toxicities) * Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to pazopanib or topotecan