This clinical trial studies the side effects and how well surgery and heated chemotherapy with or without non-heated chemotherapy after surgery works in treating patients with ovarian, fallopian tube, uterine, or peritoneal cancer. Giving a dose of heated chemotherapy into the abdomen during surgery that is done to remove ovarian, fallopian tube, uterine, or peritoneal cancer may help lower the risk of the cancer coming back. Giving unheated chemotherapy drugs directly into the abdomen after surgery may kill more tumor cells.