A constraint-free formulation of black hole thermodynamics from the field equations

In this paper, we develop a constraint-free formulation that generalizes Padmanabhan’s method for deriving the first law of black hole thermodynamics directly from the Einstein field equations. In previous studies, even for multi-horizon black holes, variations were restricted to the outer horizon by imposing an additional constraint, and the [Formula: see text] term was introduced by multiplying the field equations evaluated at the outer horizon by the corresponding volume variation [Formula: see text]. However, since general variations of the black hole parameters shift both horizons, variations at both horizons must be taken into account. To this end, we propose multiplying the horizon field equations by the entropy variation [Formula: see text] under such unconstrained variations. We show that this method remains valid even in higher-derivative theories of gravity. In addition, we find that [Formula: see text]-based variation schemes generically break down for black holes characterized by three independent parameters [Formula: see text]. By working directly in the thermodynamic state space [Formula: see text], we show that the Einstein field equations evaluated at the outer horizon can be also interpreted as the first law of black hole thermodynamics for general variations without imposing any additional constraints.