Sorafenib Tosylate With or Without Doxorubicin Hydrochloride in Treating Patients With Locally Advanced or Metastatic Liver Cancer

Inclusion Criteria: * Patients must have pathologically or cytologically proven hepatocellular carcinoma; known mixed histology (e.g. hepatocellular carcinoma plus cholangiocarcinoma) or fibrolamellar variant is not allowed * Patients must have locally advanced or metastatic disease; locally advanced disease is defined as disease deemed to be unresectable or non-eligible for transplant without distant metastases * Lesions must be accurately measurable in at least one dimension (longest diameter to be recorded) as \>= 2 cm with conventional techniques or as \>= 1 cm with spiral computed tomography (CT) scan * No prior adjuvant sorafenib or other v-RAF-1 murine leukemia viral oncogene homolog (Raf)/vascular endothelial growth factor (VEGF) inhibitors; other prior adjuvant therapy is allowed if completed \> 6 months prior to registration with documented recurrence of hepatocellular carcinoma (HCC) * Patients may have been treated with loco regional therapies provided that they either have: * A target lesion that has not been subjected to local therapy or * The target lesion(s) within the field of the local therapy has shown an increase of \>= 20% in the size since last treatment * Such therapy must be completed at least 4 weeks prior to registration; patients that have received palliative radiation therapy to the bone need not wait 4 weeks to begin protocol therapy * Prior therapies allowed include the following: * Bland embolization, radiation, radioactive microsp…