  On an $n$-dimensional, massless, topological black hole with hyperbolic
sections, we construct the two-point function both of a ground state and of a
thermal state for a real, massive, free scalar field arbitrarily coupled to
scalar curvature and endowed with Robin boundary conditions at conformal
infinity. These states are used to compute the response of an Unruh-DeWitt
detector coupled to them for an infinite proper time interval along static
trajectories. As an application, we focus on the massless conformally coupled
case and we show, numerically, that the anti-Hawking effect, which is manifest
on the three-dimensional case, does not occur if we consider a four-dimensional
massless hyperbolic black hole. On the one hand, we argue that this result is
compatible with what happens in the three- and four dimensional Minkowski
spacetime, while, on the other hand, we stress that it generalizes existing
results concerning the anti-Hawking effect on black hole spacetimes.
