  We propose a general approach for the construction of modified gravity which
is invariant under foliation-preserving diffeomorphisms. Special attention is
paid to the formulation of modified $F(R)$ Ho\v{r}ava-Lifshitz gravity (FRHL),
whose Hamiltonian structure is studied. It is demonstrated that the
spatially-flat FRW equations of FRHL are consistent with the constraint
equations. The analysis of de Sitter solutions for several versions of FRHL
indicates that the unification of the early-time inflation with the late-time
acceleration is possible. It is shown that a special choice of parameters for
FRHL leads to the same spatially-flat FRW equations as in the case of
traditional $F(R)$-gravity. Finally, an essentially most general modified
Ho\v{r}ava-Lifshitz gravity is proposed, motivated by its fully
diffeomorphism-invariant counterpart, with the restriction that the action does
not contain derivatives higher than the second order with respect to the time
coordinate.
