  In classical general relativity described by Einstein-Hilbert gravity, black
holes behave as thermodynamic objects. In particular, the laws of black hole
mechanics can be interpreted as laws of thermodynamics. The first law of black
hole mechanics extends to higher derivative theories via the Noether charge
construction of Wald. One also expects the statement of the second law, which
in Einstein-Hilbert theory owes to Hawking's area theorem, to extend to higher
derivative theories. To argue for this however one needs a notion of entropy
for dynamical black holes, which the Noether charge construction does not
provide. We propose such an entropy function for the family of Lovelock
theories, treating the higher derivative terms as perturbations to the
Einstein-Hilbert theory. Working around a dynamical black hole solution, and
making no assumptions about the amplitude of departure from equilibrium, we
construct a candidate entropy functional valid to all orders in the low energy
effective field theory. This entropy functional satisfies a second law, modulo
a certain subtle boundary term, which deserves further investigation in
non-spherically symmetric situations.
