  For a perturbation of the state of a Conformal Field Theory (CFT), the
response of the entanglement entropy is governed by the so-called "first law"
of entanglement entropy, in which the change in entanglement entropy is
proportional to the change in energy. Whether such a first law holds for other
types of perturbations, such as a change to the CFT Lagrangian, remains an open
question. We use holography to study the evolution in time $t$ of entanglement
entropy for a CFT driven by a $t$-linear source for a conserved $U(1)$ current
or marginal scalar operator. We find that although the usual first law of
entanglement entropy may be violated, a first law for the rates of change of
entanglement entropy and energy still holds. More generally, we prove that this
first law for rates holds in holography for any asymptotically
$(d+1)$-dimensional Anti-de Sitter metric perturbation whose $t$ dependence
first appears at order $z^d$ in the Fefferman-Graham expansion about the
boundary at $z=0$.
