  We compute the black hole radiation spectrum in the presence of
high-frequency dispersion in a large set of situations. In all cases, the
spectrum diverges like the inverse of the Killing frequency. When studying the
low-frequency spectrum, we find only two regimes: an adiabatic one where the
corrections with respect to the standard temperature are small, and an abrupt
one regulated by dispersion, in which the near-horizon metric can be replaced
by step functions. The transition from one regime to the other is governed by a
single parameter which also governs the net redshift undergone by dispersive
modes. These results can be used to characterize the quasiparticles spectrum of
recent and future experiments aiming to detect the analogue Hawking radiation.
They also apply to theories of quantum gravity which violate Lorentz
invariance.
