Avelumab and Trabectedin in Treating Patients With Liposarcoma or Leiomyosarcoma That is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * Must have a histologically confirmed diagnosis of advanced (metastatic or unresectable) soft tissue sarcoma with one of the following subtypes: * Leiomyosarcoma * Liposarcoma * Subject must be clinically indicated to receive trabectedin therapy as part of routine care. Subjects may be first line, or have received any number of prior systemic therapies * Total bilirubin level =\< 1.5 x the upper limit of normal (ULN) mg/dL * Aspartate aminotransferase (AST) =\< 2.5 x ULN and alanine aminotransferase (ALT) =\< 2.5 x ULN * Alkaline phosphatase \< 2.5 x ULN * Serum creatinine =\< 1.5 x ULN * Calculated creatinine clearance \>= 30 mL/min using the Cockcroft-Gault formula may be included * Creatinine phosphokinase (CPK) =\< 2.5 x ULN * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelet count \>= 100,000/mm\^3 (100 x 10\^9/L) * Hemoglobin \>= 9 g/dL * Subject must demonstrate a left ventricular ejection fraction (LVEF) \> 45% by echocardiography (ECHO) or multigated acquisition scan (MUGA) * Male or non-pregnant and non-breast feeding female: * Females of child-bearing potential must agree to use highly effective contraception without interruption from initiation of therapy and while on study medication and have a negative serum pregnancy test (beta - human chorionic gonadotropin \[hCG\]) result at screening and agree to ongoing pregnancy testing during the course of the study, and at the end of study treatment; a highly effective method of c…