  We study a version of the recently proposed modified $F(R)$
Ho\v{r}ava-Lifshitz gravity that abandons the projectability condition of the
lapse variable. We discovered that the projectable version of this theory has a
consistent Hamiltonian structure, and that the theory has interesting
cosmological solutions which can describe the eras of accelerated expansion of
the universe in a unified manner. The usual Ho\v{r}ava-Lifshitz gravity is a
special case of our theory. Hamiltonian analysis of the non-projectable theory,
however, shows that this theory has serious problems. These problems are
compared with those found in the original Ho\v{r}ava-Lifshitz gravity. A
general observation on the structure of the Poisson bracket of Hamiltonian
constraints in all theories of the Ho\v{r}ava-Lifshitz type is made: in the
resulting tertiary constraint the highest order spatial derivative of the lapse
$N$ is always of uneven order. Since the vanishing of the lapse (N=0) is
required by the preservation of the Hamiltonian constraints under time
evolution, we conclude that the non-projectable version of the theory is
physically inconsistent.
