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In previous work, a first law of generalized entropy was derived fromsemiclassical gravitational dynamics around thermal setups using an assumedrelation between the matter modular Hamiltonian and the gravitational stresstensor. Allowing for non-minimal coupling between curvature and any tensormatter fields, we show however, that the modular Hamiltonian of thermal statesis given by the integrated bulk Noether current associated to time translationplus a spacetime boundary term. One generally cannot express this in terms ofgravitational stress tensor components. Still, working with the correctexpression for the modular Hamiltonian, we are able to recover a first law ofgeneralized entropy, with added benefits over the previous result. Firstly, anyWald-Dong contributions to generalized entropy resulting from non-minimalcoupling between matter and curvature are included. Secondly, in gravitationalequations of motion, we allow for a non-vanishing stress tensor expectationvalue in the unperturbed background and state, and account for background fieldperturbations as part of its variation. Finally, the quantum matter is allowedto contribute nontrivially to asymptotic energy, e.g. as is necessary, even fora minimally coupled Maxwell field, to recover the expected thermodynamic firstlaw of charged black holes.

The generalized first law for more general matter