Thermodynamics of Rotating Charged Black Branes in Third Order Lovelock Gravity and the Counterterm Method

We generalize the quasilocal definition of the stress energy tensor of Einstein gravity to the case of third order Lovelock gravity, by introducing the surface terms that make the action well-defined. We also introduce the boundary counterterm that removes the divergences of the action and the conserved quantities of the solutions of third order Lovelock gravity with zero curvature boundary at constant $t$ and $r$. Then, we compute the charged rotating solutions of this theory in $n+1$ dimensions with a complete set of allowed rotation parameters. These charged rotating solutions present black hole solutions with two inner and outer event horizons, extreme black holes or naked singularities provided the parameters of the solutions are chosen suitable. We compute temperature, entropy, charge, electric potential, mass and angular momenta of the black hole solutions, and find that these quantities satisfy the first law of thermodynamics. We find a Smarr-type formula and perform a stability analysis by computing the heat capacity and the determinant of Hessian matrix of mass with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles, and show that the system is thermally stable. This is commensurate with the fact that there is no Hawking-Page phase transition for black objects with zero curvature horizon.