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

On Stability of Asymptotically Free Mimetic Ho\v{r}ava Gravity