  For a Kerr black hole perturbed by a particle on the "corotating" circular
orbit (angular velocity equal to that of the unperturbed event horizon), the
black hole remains in equilibrium in the sense that the perturbed event horizon
is a Killing horizon of the helical Killing field. The associated surface
gravity is constant over the horizon and should correspond to the physical
Hawking temperature. We calculate the perturbation in surface
gravity/temperature, finding it negative: the moon has a cooling effect on the
black hole. We also compute the change in horizon angular frequency, which is
positive, and the change in surface area/entropy, which vanishes.
