Testing the Anti-cancer Drug Darolutamide in Patients With Testosterone-Driven Salivary Gland Cancers

Inclusion Criteria: * Patients must have histologically or cytologically confirmed salivary gland cancer that is recurrent/metastatic or unresectable/locally advanced, with AR expression detected by immunohistochemistry (IHC) on a Clinical Laboratory Improvement Act (CLIA)-approved assay. Androgen receptor testing by immunohistochemistry (IHC) can be performed locally in a CLIA (Clinical Laboratory Improvement Amendments) certified lab * Patients must have measurable disease * Patients must have not had prior AR-targeted therapy, except for AR-targeted therapy administered in the neoadjuvant and/or adjuvant setting and with disease recurrence more than 6 months since treatment completion * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of darolutamide in combination with leuprolide acetate in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (exception: patients with elevated bilirubin due to Gilbert's disease would be eligible for the trial) * Aspartate aminotransferase (AST) (serum (glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase (\[SGPT\]) =\< 3 x institutional ULN * Creatinine =\< 1.5 x institutional ULN * Glomerular filtration rate (GFR) \>= 30 mL/min/1.73 m\^2 (by Chronic Kidney Disease Epidemiology Collaboration \[CKD-EPI\]) * 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 * 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 2B or better * The effects of darolutamide on the developing human fetus are unknown. For this reason and because androgen receptor inhibitor agents as well as other therapeutic agents used in this trial are known to be teratogenic (leuprolide-acetate), women of child-bearing potential and men must agree to use adequate contraception (non-hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. 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 and women treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 7 days after completion of darolutamide administration or after the depot interval for the leuprolide-acetate dose used has been completed, whichever is longer * Ability to understand and the willingness to sign a written informed consent document * Patients must have tumors that are safely accessible for biopsy. * Note: Two research biopsies are mandated in this trial. If the biopsy is deemed to be unsafe after attempting the first biopsy, the patient will remain eligible for the trial and subsequent tumor biopsies will not be required 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 and peripheral neuropathy * Patients with a vascular or ischemic event within 6 months of study registration * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to darolutamide or leuprolide acetate * Patients on combined P-gp and strong or moderate CYP3A inducers or BCRP substrates are excluded. 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 * Patients with uncontrolled intercurrent illness * Pregnant women are excluded from this study because darolutamide is an androgen receptor 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 darolutamide and leuprolide-acetate, breastfeeding should be discontinued if the mother is treated with darolutamide and leuprolide-acetate. These potential risks may also apply to other agents used in this study. * Patients with moderate hepatic impairment (Child-Pugh Class B or C)