  We consider a variant of the Nojiri-Odintsov covariant Horava-like
gravitational model, where diffeomorphism invariance is broken dynamically via
a non-standard coupling to a perfect fluid. The theory allows to address some
of the potential instability problems present in Horava-Lifshitz gravity due to
explicit diffeomorphism invariance breaking. The fluid is instead constructed
from a scalar field constrained by a Lagrange multiplier. In fact, the Lagrange
multiplier construction allows for an extension of the Horava-like model to
include the scalar field of mimetic gravity, an extension which we thoroughly
explore. By adding a potential for the scalar field, we show how one can
reproduce a number of interesting cosmological scenarios. We then turn to the
study of perturbations around a flat FLRW background, showing that the fluid in
question behaves as an irrotational fluid, with zero sound speed. To address
this problem, we consider a modified version of the theory, adding higher
derivative terms in a way which brings us beyond the Horndeski framework. We
compute the sound speed in this modified higher order mimetic Horava-like model
and show that it is non-zero, which means that perturbations therein can be
sensibly defined. Caveats to our analysis, as well as comparisons to
projectable Horava-Lifshitz gravity, are also discussed. In conclusion, we
present a theory of gravity which preserves diffeomorphism invariance at the
level of the action but breaks it dynamically in the UV, reduces to General
Relativity in the IR, allows the realization of a number of interesting
cosmological scenarios, is well defined when considering perturbations around a
flat FLRW background, and features cosmological dark matter emerging as an
integration constant.
