This phase I trial studies the side effects and best dose of pazopanib hydrochloride (pazopanib) when given together with cetuximab in treating patients with incurable recurrent or metastatic head and neck cancer. Pazopanib may stop the growth of cancer by blocking blood flow to the tumor. Pazopanib may also block some of the enzymes needed for cell growth. Cetuximab is a monoclonal antibody that blocks the ability of some tumor cells to grow and spread. Giving pazopanib with cetuximab may provide a more effective treatment for patients with advanced head and neck cancer.