  Asymptotically free mimetic gravity has been introduced as a proposal for a
classical limiting curvature theory with the purpose of singularity resolution.
It was found that in a spatially flat universe an initial stage of exponential
expansion with graceful exit is a generic consequence, regardless of the matter
content. In this work I will analyze linear stability of cosmological
perturbations in such a model, considering only the degrees of freedom of pure
mimetic gravity. I show that the addition of Ho\v{r}ava-gravity-like higher
order spatial curvature terms can lift the gradient instability of scalar
perturbations, even when the gradient term has the wrong sign throughout.
Calculating the primordial spectra of tensor and scalar perturbations in the
simplest single component model, I find that the initially scale invariant
spectra turn out to be destroyed later by the rapidly varying speed of sound at
horizon exit.
