  We review our recent results on the renormalization procedure for a free
quantum scalar field with modified dispersion relations in curved spacetimes.
For dispersion relations containing up to $2s$ powers of the spatial momentum,
the subtraction necessary to renormalize $<\phi^2>$ and $<T_{\mu\nu}>$ depends
on $s$. We first describe our previous analysis for spatially flat
Friedman-Robertson-Walker and Bianchi type I metrics. Then we present a new
power counting analysis for general background metrics in the weak field
approximation.
