Towards metric-like higher spin gauge theories in three dimensions

We consider the coupling of a symmetric spin-3 gauge field phi(mu nu rho) to three-dimensional gravity in a second-order metric-like formulation. The action that corresponds to an SL(3, R) x SL(3, R) Chern-Simons theory in the frame-like formulation is identified to quadratic order in the spin-3 field. We apply our result to compute corrections to the area law for higher spin black holes using Wald's entropy formula.