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We use the Iyer-Wald formalism to derive an extended first law ofentanglement that includes variations in the cosmological constant, Newton'sconstant and --in the case of higher-derivative theories-- all the additionalcouplings of the theory. In Einstein gravity, where the number of degrees offreedom $N^2$ of the dual field theory is a function of $\Lambda$ and $G$, ourapproach allows us to vary $N$ keeping the field theory scale fixed or to varythe field theory scale keeping $N$ fixed. We also derive an extended first lawof entanglement for Gauss-Bonnet and Lovelock gravity and show that in thesecases all the extra variations reorganize nicely in terms of the centralcharges of the theory. Finally, we comment on the implications forrenormalization group flows and c-theorems in higher dimensions.

Holographic entanglement chemistry