$g_{tt}{g}_{rr}=-1$ black hole thermodynamics in extended quasi-topological gravity

We present a unified framework for the discussion of black hole thermodynamics of $d$-dimensional static black holes with spherical, toroidal or compact hyperbolic horizon topology satisfying $g_{tt}{g}_{rr}=-1$ in Schwarzschild gauge. To that end, we consider any such black hole as a solution to an integrable $2$-dimensional effective dilaton theory and thereby as a vacuum solution to an extended notion of $d$-dimensional quasi-topological gravity. We show that the generating function determining $f(r)=-g_{tt}$ in the integrated equation of motion provides the thermodynamic mass in a generalised first law with entropy computed as the Wald entropy. The framework presented here can be applied to singular and regular black holes with flat or anti-de Sitter asymptotics.