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'>δaδa of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension δa*'>δa∗δa* to include odd dimensional boundaries, times the ratio S/a*'>S/a∗S/a* . Since a*'>a∗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'>μδNμδN , where the chemical potential μ is given by the entanglement entropy per degree of freedom.