  We analyze the semiclassical Einstein equations for quantum scalar fields
satisfying modified dispersion relations. We first discuss in detail the
renormalization procedure based on adiabatic subtraction and dimensional
regularization. We show that, contrary to what expected from power counting
arguments, in 3+1 dimensions the subtraction involves up to the fourth
adiabatic order even for dispersion relations containing higher powers of the
momentum. Then we analyze the dependence of the trace of the renormalized
energy momentum tensor with the scale of new physics, and we recover the usual
trace anomaly in the appropriate limit. We also find selfconsistent de Sitter
solutions for dispersion relations that contain up to the fourth power of the
momentum. Using this particular example, we also discuss the possibility that
the modified dispersion relation can be mimicked at lower energies by an
effective initial state in a theory with the usual dispersion relation.
