This randomized phase III trial is studying sorafenib tosylate and doxorubicin hydrochloride to see how well they work compared with sorafenib tosylate alone in treating patients with locally advanced or metastatic liver cancer. Sorafenib tosylate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth and by blocking blood flow to the tumor. Drugs used in chemotherapy, such as doxorubicin hydrochloride, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. It is not yet known whether giving sorafenib tosylate together with doxorubicin hydrochloride is more effective than sorafenib tosylate alone in treating liver cancer.