  For time-independent excited states in conformal field theories, the
entanglement entropy of small subsystems satisfies a `first law'-like relation,
in which the change in entanglement is proportional to the energy within the
entangling region. Such a law holds for time-dependent scenarios as long as the
state is perturbatively close to the vacuum, but is not expected otherwise. In
this paper we use holography to investigate the spread of entanglement entropy
for unitary evolutions of special physical interest, the so-called global
quenches. We model these using AdS-Vaidya geometries. We find that the first
law of entanglement is replaced by a linear response relation, in which the
energy density takes the role of the source and is integrated against a
time-dependent kernel with compact support. For adiabatic quenches the standard
first law is recovered, while for rapid quenches the linear response includes
an extra term that encodes the process of thermalization. This extra term has
properties that resemble a time-dependent `relative entropy'. We propose that
this quantity serves as a useful order parameter to characterize
far-from-equilibrium excited states. We illustrate our findings with concrete
examples, including generic power-law and periodically driven quenches.
