 This partially randomized phase III trial studies the side effects and how well sirolimus works when given together with docetaxel and carboplatin in treating patients with hormone-resistant prostate cancer that has spread to other places in the body. Biological therapies, such as sirolimus, use substances made from living organisms that may stimulate or suppress the immune system in different ways and stop tumor cells from growing. Drugs used in chemotherapy, such as docetaxel and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving sirolimus together with docetaxel and carboplatin may kill more tumor cells. 