  This paper presents the first derivation of the quadratic action for
curvature perturbations, $\zeta$, within the framework of cuscuton gravity. We
study the scalar cosmological perturbations sourced by a canonical single
scalar field in the presence of cuscuton field. We identify $\zeta$ as comoving
curvature with respect to the source field and we show that it retains its
conservation characteristic on super horizon scales. The result provides an
explicit proof that cuscuton modification of gravity around
Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is ghost free. We also
investigate the potential development of other instabilities in cuscuton
models. We find that in a large class of these models, there is no generic
instability problem. However, depending on the details of slow-roll parameters,
specific models may display gradient instabilities.
