Smarr formula and an extended first law for Lovelock gravity

Abstract

We study properties of static, asymptotically AdS black holes in Lovelock gravity. Our main result is a Smarr formula that gives the mass in terms of geometrical quantities together with the parameters of the Lovelock theory. As in Einstein gravity, the Smarr formula follows from applying the first law to an infinitesimal change in the overall length scale. However, because the Lovelock couplings are dimensionful, we must first prove an extension of the first law that includes their variations. Key ingredients in this construction are the Killing–Lovelock potentials associated with each of the higher curvature Lovelock interactions. Geometric expressions are obtained for the new thermodynamic potentials conjugate to the variation of the Lovelock couplings.