  We carefully study the implications of adiabaticity for the behavior of
cosmological perturbations. There are essentially three similar but different
definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid
$\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and
the last one is valid only on superhorizon scales. The first two definitions
coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the
perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the
general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse
function in the comoving sli\-cing $A_c$ and $\delta P_{nad}$ valid for
arbitrary matter field in any theory of gravity, by using only momentum
conservation. The relation implies that as long as $c_s\neq c_w$, the uniform
density, comoving and the proper-time slicings coincide approximately for any
gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In
the case of general relativity this gives the equivalence between the comoving
curvature perturbation $R_c$ and the uniform density curvature perturbation
$\zeta$ on superhorizon scales, and their conservation.
  We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta
P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer
holds. Namely we consider the so-called ultra slow-roll inflation. In this case
both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we
find that it is crucial to take into account the next-to-leading order term in
$\zeta$'s spatial gradient expansion to show its non-conservation, even on
superhorizon scales. This is an example of the fact that adiabaticity (in the
thermodynamic sense) is not always enough to ensure the conservation of $R_c$
or $\zeta$.
