  We propose the most general modified first-order Ho\v{r}ava-Lifshitz (HL)
gravity, whose action does not contain time derivatives higher than the second
order. The Hamiltonian structure of this theory is studied in all the details
in the case of the spatially-flat FRW space-time, demonstrating many of the
features of the general theory. It is shown that, with some plausible
assumptions, including the projectability of the lapse function, this model is
consistent. As a large class of such theories, the modified HL $F(R)$ gravity
is introduced. The study of its ultraviolet properties shows that its $z=3$
version seems to be renormalizable in the same way as the original HL proposal.
The Hamiltonian analysis of the modified HL $F(R)$ gravity shows that it is in
general a consistent theory. The $F(R)$ gravity action is also studied in the
fixed-gauge form, where the appearance of a scalar field is particularly
illustrative. Then the spatially-flat FRW cosmology for this $F(R)$ gravity is
investigated. It is shown that a special choice of parameters for this theory
leads to the same equations of motion as in the case of traditional $F(R)$
gravity. Nevertheless, the cosmological structure of the modified HL $F(R)$
gravity turns out to be much richer than for its traditional counterpart. The
emergence of multiple de Sitter solutions indicates to the possibility of
unification of early-time inflation with late-time acceleration within the same
model. Power-law $F(R)$ theories are investigated in detail. It is analytically
shown that they have a quite rich cosmological structure: early/late-time
cosmic acceleration of quintessence, as well as of phantom types. Also it is
demonstrated that all the four known types of finite-time future singularities
may occur in the power-law HL $F(R)$ gravity. Finally, a covariant proposal for
(renormalizable) $F(R)$ gravity within the HL spirit is presented.
