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 δa of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension δa∗ 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 μδN, where the chemical potential μ is given by the entanglement entropy per degree of freedom.