  We give an account of the physical behaviour of a quasiparticle horizon due
to non-Lorentz invariant modifications of the effective space-time experienced
by the quasiparticles (``matter'') for high momenta. By introducing a
``relativistic'' conserved energy-momentum tensor, we derive quasi-equilibrium
states of the fluid across the ``Landau'' quasiparticle horizon at temperatures
well above the quantum Hawking temperature. Nonlinear dispersion of the
quasiparticle energy spectrum is instrumental for quasiparticle communication
and exchange across the horizon. It is responsible for the establishment of the
local thermal equilibrium across the horizon with the Tolman temperature being
inhomogeneous behind the horizon. The inhomogeneity causes relaxation of the
quasi-equilibrium states due to scattering of thermal quasiparticles, which
finally leads to a shrinking black hole horizon. This process serves as the
classical thermal counterpart of the quantum effect of Hawking radiation and
will allow for an observation of the properties of the horizon at temperatures
well above the Hawking temperature. We discuss the thermal entropy related to
the horizon. We find that only the first nonlinear correction to the energy
spectrum is important for the thermal properties of the horizon. They are fully
determined by an energy of order $E_{\rm Planck}(T/E_{\rm Planck})^{1/3}$,
which is well below the Planck energy scale $E_{\rm Planck}$, so that Planck
scale physics is not involved in determining thermal quantities related to the
horizon.
