Thermodynamical first laws of black holes in quadratically-extended gravities

Einstein gravities in general dimensions coupled to a cosmological constant and extended with quadratic curvature invariants admit a variety of black holes that may asymptote to Minkowski, anti–de Sitter or Lifshitz spacetimes. We adopt the Wald formalism to derive an explicit formula for calculating the thermodynamical first law for the static black holes with spherical, toric, or hyperbolic isometries in these theories. This allows us to derive or rederive the first laws for a wide range of black holes in the literature. Furthermore, we construct many new exact solutions and obtain their first laws.