  In [arXiv:2105.06455, arXiv:2206.04538], the authors have been able to argue
for an ultra-local version of the second law of black hole mechanics, for
arbitrary diffeomorphism invariant theories of gravity non-minimally coupled to
matter fields, by constructing an entropy current on the dynamical horizon with
manifestly positive divergence. This has been achieved by working in the
horizon-adapted coordinate system. In this work, we show that the local entropy
production through the divergence of the entropy current is covariant under
affine reparametrizations that leave the gauge of horizon-adapted coordinates
invariant. We explicitly derive a formula for how the entropy current
transforms under such coordinate transformations. This extends the analysis of
[arXiv:2204.08447] for arbitrary diffeomorphism invariant theories of gravity
non-minimally coupled to matter fields. We also study the Iyer-Wald ambiguities
of the covariant phase formalism that generically plague the components of the
entropy current.
