Extended First Law for Entanglement Entropy in Lovelock Gravity

The first law for the holographic entanglement entropy of spheres in a boundary CFT with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation $\delta a$ of the A-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension $\delta a^*$ to include odd dimensional boundaries, times the ratio $S/a^*$. Since $a^*$ is a measure of the number of degrees of freedom $N$ per unit volume of the boundary CFT, this new term has the form $\mu\delta N$, where the chemical potential $\mu$ is given by the entanglement entropy per degree of freedom.