  Among all the different techniques to derive the Hawking effect, the approach
based on gravitational anomaly by Robinson and Wilczek provides a simple and
satisfactory origin of the black hole radiation. In this picture, the effective
near horizon physics becomes chiral and contains gravitational anomaly.
Nevertheless, the underlying description must be generally covariant, and
therefore we require a compensating energy-momentum flux whose divergence
cancels the anomaly at the horizon. Remarkably, the energy flux associated with
the Hawking emission from the horizon exactly cancels the gravitational anomaly
and restores the general covariance at the quantum level. In this work, we
present a generalization of the original derivation for a stationary
axisymmetric black hole solution of any gravity theory which differs
perturbatively from general relativity. The crucial input of the calculation is
a remarkable simplification of the near horizon geometry and the validity of
the zeroth law of black hole mechanics.
