  When Lorentz invariance is violated at high energy, the laws of black hole
thermodynamics are apparently no longer satisfied. To shed light on this
observation, we study dispersive fields in de Sitter space. We show that the
Bunch-Davies vacuum state restricted to the static patch is no longer thermal,
and that the Tolman law is violated. However we also show that, for free fields
at least, this vacuum is the only stationary stable state, as if it were in
equilibrium. We then present a precise correspondence between dispersive
effects found in de Sitter and in black hole metrics. This indicates that the
consequences of dispersion on thermodynamical laws could also be similar.
