We calculate parameters in the low energy gravitational effective action and the entanglement entropy in a wide class of theories characterized by improved ultraviolet (UV) behavior. These include i) local and non-local Lorentz invariant theories in which inverse propagator is modified by higher-derivative terms and ii) theories described by non-Lorentz invariant Lifshitz type field operators. We demonstrate that the induced cosmological constant, gravitational couplings and the entropy are sensitive to the way the theory is modified in UV. For non-Lorentz invariant theories the induced gravitational effective action is of the Horava-Lifshitz type. We show that under certain conditions imposed on the dimension of the Lifshitz operator the couplings of the extrinsic curvature terms in the effective action are UV finite. Throughout the paper we systematically exploit the heat kernel method appropriately generalized for the class of theories under consideration.

The Hamiltonian formalism of the generalized unimodular gravity theory, which was recently suggested as a model of dark energy, is shown to be a complicated example of constrained dynamical system. The set of its canonical constraints has a bifurcation -splitting of the theory into two branches differing by the number and type of these constraints, one of the branches effectively describing a gravitating perfect fluid with the time-dependent equation of state, which can potentially play the role of dark energy in cosmology. The first class constraints in this branch generate local gauge symmetries of the Lagrangian action -two spatial diffeomorphisms -and rule out the temporal diffeomorphism which does not have a realization in the form of the canonical transformation on phase space of the theory and turns out to be either nonlocal in time or violating boundary conditions at spatial infinity. As a consequence, the Hamiltonian reduction of the model enlarges its physical sector from two general relativistic modes to three degrees of freedom including the scalar graviton. This scalar mode is free from ghost and gradient instabilities on the Friedmann background in a wide class of models subject to a certain restriction on time-dependent parameter w of the dark fluid equation of state, p = wε. For a special family of models this scalar mode can be ruled out even below the phantom divide line w = −1, but this line cannot be crossed in the course of the cosmological expansion. This is likely to disable the generalized unimodular gravity as a model of the phenomenologically consistent dark energy scenario, but opens the prospects in inflation theory with a scalar graviton playing the role of inflaton.