  We investigate the Hamiltonian structure of linearized extended Ho\v{r}ava-
Lifshitz gravity in a flat cosmological background following the
Faddeev-Jackiw's Hamiltonian reduction formalism. The Hamiltonian structure of
extended Ho\v{r}ava-Lifshitz gravity is similar to that of the projectable
version of original Ho\v{r}ava-Lifshitz gravity, in which there is one primary
constraint and so there are two physical degrees of freedom. We also find that
extra scalar graviton mode in an inflationary background can be decoupled from
the matter field in the infrared (IR) limit, but it is coupled to the matter
field in a general cosmological background. But it is necessary to go beyond
linear order in order to draw any conclusion of the strong coupling problem.
