The covariant action of higher spin black holes in three dimensions

We propose a set of boundary terms for higher spin theories in ads3 that lead to a well-defined variational principle compatible with dirichlet boundary conditions for the metric and higher spin fields. these boundary terms are valid for higher spin theories in the fefferman-graham gauge and they allow us to compute the canonical free energy of higher spin black holes directly from the euclidean, covariant, on-shell action. using these results we reproduce the thermodynamics of the higher spin black hole of ammon, gutperle, kraus, and perlmutter and comment on the corresponding theory of induced $$ mathcal{w} $$ -gravity at the boundary.