Testing the Anti-cancer Drug, Rogaratinib (BAY 1163877), for Treatment of Advanced Sarcoma With Alteration in Fibroblast Growth Factor Receptor (FGFR 1-4), and in Patients With SDH-deficient Gastrointestinal Stromal Tumor (GIST)

Inclusion Criteria: * Participant must have histologically confirmed sarcoma with FGFR alteration identified by next-generation sequencing profiling with the exception of SDH-deficient GIST who can be enrolled regardless of FGFR status. Initial testing can be performed on archival tissue, if available. Patients must have locally advanced or metastatic disease that is not amenable to surgery * Presence of measurable disease: Patient must have measurable disease * Patients must have progressed following at least one standard prior chemotherapy regimen with the exception of SDH-deficient GIST for which there is no standard of care * Participant must be willing to undergo pre-treatment biopsy if disease site is amenable to biopsy and low risk for the biopsy procedure. If biopsy is not possible, eligibility may be approved after discussion with the Study Chair. Of note, a minimum of 15 participants in each arm open to stage 2 should have disease amenable to biopsy. For those arms open in stage 1, all patients should have biopsiable disease. * Age \>= 18 years * Because no dosing or adverse event data are currently available on the use of rogaratinib (BAY 1163877) in patients \<18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Hemoglobin \>= 8.0 g/dL * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 3.0 x institutional ULN (unless liver metastases are present in which case it must be =\< 5 x ULN) * Glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 (using the Chronic Kidney Disease Epidemiology Collaboration \[CKD-EPI\] formula) * Human immunodeficiency virus (HIV)-infected patients on effective non-CYP3A4 interacting 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 must be disease-free of prior invasive malignancies for \> 5 years with the exception of curatively-treated basal cell or squamous cell carcinoma of the skin or carcinoma in situ of the cervix * NOTE: If there is a history of prior malignancy, patients must not be receiving other specific treatment for that cancer * Patients should have completed prior treatment for their cancer: chemotherapy or radiotherapy must have been completed for greater than 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients should have recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia * 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 2B or better * Patients must have a QTc interval length of below 450 millisecond (msec) * Participant is willing to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up * Participant must be able to swallow and maintain pills * Women of childbearing potential must have a negative urine or serum pregnancy test within 28 days of initial dose of rogaratinib (BAY 1163877), and again within 7 days prior to treatment on day 1. If screening occurs within 7 days of day 1, only one pregnancy test is required * The effects of rogaratinib (BAY 1163877) on the developing human fetus are unknown. For this reason and because kinase inhibitor agents are known to be teratogenic, 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 4 months after completion of rogaratinib (BAY 1163877). 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. 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 rogaratinib (BAY 1163877) administration * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to rogaratinib (BAY 1163877) * Concomitant administration with sensitive substrates/narrow therapeutic index drugs of CYP3A4, P-gp BCRP, MATE1, and MATE2K, and strong inhibitors and inducers of CYP3A4 should be avoided. Use caution with strong inhibitors and inducers of P-glycoprotein (gp). Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. 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 * Concomitant administration of medications that prolong QT/QTc interval is prohibited in accordance with the published FDA guidance "E14 Clinical Evaluation of QT/QTc Interval Prolongation and Proarrhythmic Potential for Non-Antiarrhythmic Drugs". * Patients with disturbed calcium and/or phosphate metabolism are excluded from this study * Patients with uncontrolled intercurrent illness * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because rogaratinib (BAY 1163877) is kinase inhibitor agent 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 rogaratinib (BAY 1163877), breastfeeding should be discontinued if the mother is treated with rogaratinib (BAY 1163877)