Extended First Law for Entanglement Entropy in Lovelock Gravity

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) 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.