We use the Iyer-Wald formalism to derive an extended first law of entanglement that includes variations in the cosmological constant, Newton’s constant and—in the case of higher-derivative theories—all the additional couplings of the theory. In Einstein gravity, where the number of degrees of freedom $N^2$ of the dual field theory is a function of $\mathrm{\Lambda }$ and $G$, our approach allows us to vary $N$ by keeping the field theory scale fixed or to vary the field theory scale by keeping $N$ fixed. We also derive an extended first law of entanglement for Gauss-Bonnet and Lovelock gravity and show that in these cases all the extra variations reorganize nicely in terms of the central charges of the theory. Finally, we comment on the implications for renormalization group flows and $c$-theorems in higher dimensions.

• Received 14 March 2017

DOI:https://doi.org/10.1103/PhysRevD.95.106015