  We calculate the energy and angular momentum fluxes across the event horizon
of a tidally deformed, rapidly rotating black hole to next-to-leading order in
the curvature of the external spacetime. These are expressed in terms of tidal
quadrupole moments and their time derivatives, which provide a characterization
of a generic tidal environment. As an application of our results, we provide an
expression for the energy and angular-momentum fluxes across the horizon when
the black hole is a member of a binary system on a slowly-moving,
quasi-circular orbit. Our expressions are accurate to 1.5 post-Newtonian order
beyond the leading-order fluxes, but they are valid for arbitrary mass ratios.
We compare our results to those previously obtained in the case of an extreme
mass ratio binary, and find that they do not agree at the 1.5 post-Newtonian
order. We investigate a number of possible sources for this discrepancy, but
are ultimately unable to resolve it.
