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The first law for the holographic entanglement entropy of spheres in aboundary CFT with a bulk Lovelock dual is extended to include variations of thebulk Lovelock coupling constants. Such variations in the bulk correspond toperturbations within a family of boundary CFTs. The new contribution to thefirst law is found to be the product of the variation $\delta a$ of the A-typetrace anomaly coefficient for even dimensional CFTs, or more generally itsextension $\delta a^*$ to include odd dimensional boundaries, times the ratio$S/a^*$. Since $a^*$ is a measure of the number of degrees of freedom $N$ perunit 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 perdegree of freedom.

Extended First Law for Entanglement Entropy in Lovelock Gravity