 This randomized phase IIIII trial studies how well pegylated liposomal doxorubicin hydrochloride with atezolizumab andor bevacizumab work in treating patients with ovarian, fallopian tube, or primary peritoneal cancer that has come back. Drugs used in chemotherapy, such as pegylated liposomal doxorubicin hydrochloride, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. block tumor growth Monoclonal antibodies, such as atezolizumab and bevacizumab, may interfere with the ability of tumor cells to grow and spread. It is not yet known which combination will work better in treating patients with ovarian, fallopian tube, or primary peritoneal cancer. 