  Horava and Melby-Thompson recently proposed a new version of the
Horava-Lifshitz theory of gravity, in which the spin-0 graviton is eliminated
by introducing a Newtonian pre-potential $\phi$ and a local U(1) gauge field
$A$. In this paper, we first derive the corresponding Hamiltonian,
super-momentum constraints, the dynamical equations, and the equations for
$\phi$ and $A$, in the presence of matter fields. Then, we apply the theory to
cosmology, and obtain the modified Friedmann equation and the conservation law
of energy, in addition to the equations for $\phi$ and $A$. When the spatial
curvature is different from zero, terms behaving like dark radiation and
stiff-fluid exist, from which, among other possibilities, bouncing universe can
be constructed. We also study linear perturbations of the FRW universe with any
given spatial curvature $k$, and derive the most general formulas for scalar
perturbations. The vector and tensor perturbations are the same as those
recently given by one of the present authors [A. Wang, Phys. Rev. D{\bf 82},
124063 (2010)] in the setup of Sotiriou, Visser and Weinfurtner. Applying these
formulas to the Minkowski background, we have shown explicitly that the scalar
and vector perturbations of the metric indeed vanish, and the only remaining
modes are the massless spin-2 gravitons.
