  Using AdS/CFT an extended first law of entanglement has been previously
derived for the vacuum reduced to a ball in Minkowski. The statement not only
includes perturbations of the state but also of the conformal field theory
(CFT), via variations of the generalized central charge. We clarify some
subtleties previously overlooked and use simple arguments to generalize prior
derivations to arbitrary gravity theories in the bulk as well as new regions in
the boundary CFT. Our construction also applies to two-dimensional bulk
theories and admits an interesting extension for a three-dimensional bulk,
providing a curious result regarding the thermodynamic volume in extended black
hole thermodynamics. We discuss future prospects regarding the extended first
law of entanglement.

High Energy Physics - Theory;General Relativity and Quantum Cosmology