  We study the first law for non-stationary perturbations of a stationary black
hole whose event horizon is a Killing horizon, that relates the first-order
change in the mass and angular momentum to the change in the entropy of an
arbitrary horizon cross-section. Recently, Hollands, Wald and Zhang [1] have
shown that the dynamical black hole entropy that satisfies this first law, for
general relativity, is $S_{\text{dyn}}=(1-v\partial_v)S_{\text{BH}}$, where $v$
is the affine parameter of the null horizon generators and $S_{\text{BH}}$ is
the Bekenstein-Hawking entropy, and for general diffeomorphism covariant
theories of gravity $S_{\text{dyn}}=(1-v\partial_v)S_{\text{Wall}}$, where
$S_{\text{Wall}}$ is the Wall entropy. They obtained the first law by applying
the Noether charge method to non-stationary perturbations and arbitrary
cross-sections. In this formalism, the dynamical black hole entropy is defined
as an "improved" Noether charge, which is unambiguous to first order in the
perturbation. In the present article we provide a pedagogical derivation of the
physical process version of the non-stationary first law for general relativity
by integrating the linearised Raychaudhuri equation between two arbitrary
horizon cross-sections. Moreover, we generalise the derivation of the first law
in [1] to non-minimally coupled matter fields, using boost weight arguments
rather than Killing field arguments, and we relax some of the gauge conditions
on the perturbations by allowing for non-zero variations of the horizon Killing
field and surface gravity. Finally, for $f(\text{Riemann})$ theories of gravity
we show explicitly using Gaussian null coordinates that the improved Noether
charge is $S_{\text{dyn}}=(1-v\partial_v)S_{\text{Wall}}$, which is a
non-trivial check of [1].
