The Bekenstein-Hawking entropy of black holes in Einstein’s theory of gravity is equal to a quarter of the horizon area in units of Newton’s constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of a specific metric perturbation polarization on the horizon. We present several explicit examples of static spherically symmetric black holes.

We show that the equations of motion of generalized theories of gravity are equivalent to the thermodynamic relation $\delta Q=T\delta S$. Our proof relies on extending previous arguments by using a more general definition of the Noether charge entropy. We have thus completed the implementation of Jacobson’s proposal to express Einstein’s equations as a thermodynamic equation of state. Additionally, we find that the Noether charge entropy obeys the second law of thermodynamics if the energy-momentum tensor obeys the null energy condition. Our results support the idea that gravitation on a macroscopic scale is a manifestation of the thermodynamics of the vacuum.

We discuss the possibility of entanglement islands in cosmological spacetimes with a general perfect fluid with an equation of state w. We find that flat universes with time-symmetric slices where the Hubble parameter vanishes always have islands on that slice. We then move away from such slices, considering still universes with a general perfect fluid. Under the local thermal equilibrium assumption, the comoving entropy density s_c is constant. As a result, the conditions for an island become an inequality between the energy density (or Hubble parameter) and the temperature at some time of normalization. The consequences are that islands can exist for practically all fluids that are not radiation, i.e. w ≠ 1/3. We also discuss the ramifications of our results for universes with spatial curvature. Finally, we show that islands occur in the Simple Harmonic Universe model which has no classical singularity at the background level, in contrast to all previous examples where islands occurred only in space-times with singularities.

We evaluate the Wald Noether charge entropy for a black hole in generalized theories of gravity. Expanding the Lagrangian to second order in gravitational perturbations, we show that contributions to the entropy density originate only from the coefficients of two-derivative terms. The same considerations are extended to include matter fields and to show that arbitrary powers of matter fields and their symmetrized covariant derivatives cannot contribute to the entropy density. We also explain how to use the linearized gravitational field equation rather than quadratic actions to obtain the same results. Several explicit examples are presented that allow us to clarify subtle points in the derivation and application of our method.

The Euclidean opening angle at the $r-t_E$ surface, ${\mathrm{\Theta }}_{r-t_E}$ at the horizon of a black hole, is canonically conjugate to the black hole entropy. We prove that for a $D1D5$ black hole there exists in addition to this pair, another canonical pair: the opening angle at the $r-y$ surface, ${\mathrm{\Theta }}_{r-y}$, and a Wald-like term $S_{Wr-y}$. This leads to an uncertainty at ${\mathrm{\Theta }}_{r-y}$ which suggests that the surface $r-y$ is actually a superposition of surfaces with different conical singularities. This corresponds to the same type of singularities obtained by string theory excitations of a $D1D5$ black hole.