  The cuscuton was introduced in the context of cosmology as a field with
infinite speed of propagation. It has been claimed to resemble Ho\v{r}ava
gravity in a certain limit, and it is a good candidate for an ether theory in
which a time-dependent cosmological constant appears naturally. The analysis of
its properties is usually performed in the Lagrangian framework, which makes
issues like the counting of its dynamical degrees of freedom less clear-cut.
Here we perform a Hamiltonian analysis of the theory. We show that the
homogeneous limit with local degrees of freedom has singular behavior in the
Hamiltonian framework. In other frames, it has an extra scalar degree of
freedom. The homogeneous field has regular behavior only if defined a priori as
a spatially constant field in a CMC foliation and contributing with a single
global degree of freedom. Lastly, we find conditions on the cuscuton potential
for the resulting lapse function to be non-zero throughout evolution.
