This phase I trial studies the side effects and the best dose of veliparib when given together with pegylated liposomal doxorubicin hydrochloride, carboplatin, and bevacizumab in treating patients with ovarian cancer, primary peritoneal cancer, or fallopian tube cancer that has returned after previous treatment. Veliparib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as pegylated liposomal doxorubicin hydrochloride and carboplatin, may stop the growth of tumor cells by, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Monoclonal antibodies, such as bevacizumab, can block tumor growth by blocking the ability of tumor cells to grow and spread. Bevacizumab may also stop the growth of tumors by blocking the growth of new blood vessels necessary for tumor growth. Giving veliparib together with pegylated liposomal doxorubicin hydrochloride, carboplatin, and bevacizumab may kill more tumor cells.