Lorentz non-invariance, locality, unitarity and renormalizability in quantum gravity

We discuss problems of covariant renormalization of Lorentz non-invariant, local and unitary Ho.rava gravity models in arbitrary spacetime dimensions, including perturbatively renormalizable projectable Ho.rava gravity in (2 + 1) and (3 + 1) dimensions. Renormalization group flow with asymptotically free UV fixed point is presented in (2 + 1)-dimensional theory. A related model of generalized unimodular gravity (GUMG) is shown to have interesting applications in cosmological inflation theory and within the cosmological dark energy problem. The unification of these two models sharing in common Lorentz symmetry violation, fixation of the lapse function and a number of other features is conjectured in the framework of generalized renormalization group approach.