Entanglement entropy of black holes from the first law of thermodynamics

We compute the entanglement entropy of quantum matter in spacetimes with a Killing horizon entirely on-shell, by integrating the first law of thermodynamics along a one-parameter family of equilibrium backgrounds labelled by the Hawking temperature. This yields a closed expression in terms of the renormalized stress-energy tensor and the Killing vector, finite by construction, with no replicas and no conical singularities. Its covariant form agrees with the matter terms of the Bardeen–Carter–Hawking mass variation formula. We show in details that for static patch of de Sitter space and for the non-rotating BTZ black hole it reproduces the conical entropy exactly, contact term $Q$ included.