  We consider a pseudoscalar axion-like field coupled to a Chern-Simons
gravitational anomaly term. The axion field backreacts on a rotating Kerr black
hole background, resulting in modifications in the spacetime. In an attempt to
determine potentially observable signatures, we study the angular momentum of
the system of the modified Kerr-like black hole and the axionic matter outside
the horizon of the black hole. As the strength of the coupling of the axion
field to the Chern-Simons term is increasing, the requirement that the total
angular momentum of the system remain constant forces the black hole angular
momentum to decrease. There exists a critical value of this coupling beyond
which the black hole starts to rotate in the opposite direction, with an
increasing magnitude of its angular momentum. We interpret this effect as a
consequence of the exchange of energy between the axionic matter and the
gravitational anomaly, which is sourced by the rotating black hole.
