Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity

Recently, it was discovered that lower-dimensional versions of Lovelock gravity exist as scalar-tensor theories that are examples of Horndeski gravity. We study the thermodynamics of the static black hole solutions in these theories up to cubic order through Euclidean methods. Considering solutions with spherical, planar and hyperbolic event horizons ($k=+1, 0, -1$), we show that the universality of the thermodynamics for planar black holes ($k=0$) and the extended 1st law that include the variation of the couplings together with their associated potentials hold also in lower dimensions. We find that in $D=4, 6$ where the 2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively, the Smarr relation is modified since the entropy is not a homogenous function in these dimensions. We also present a derivation of the Smarr relation and its modified version based on the global scaling properties of the reduced action that is used to obtain the solutions consistently. Unlike the other hairy black hole solutions that have been analyzed before, despite the terms in the reduced action that break the scaling symmetry, the derivation still follows from a conserved Noether charge.