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 $\mathcal{R}_c$
and the uniform density curvature perturbation $\zeta$
on superhorizon scales, and their conservation.
This is realized on superhorizon scales in standard slow-roll inflation.

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 $\mathcal{R}_c$ and $\zeta$ no longer holds.
Namely we consider the so-called ultra slow-roll inflation.
In this case both $\mathcal{R}_c$ 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 $\mathcal{R}_c$ or $\zeta$.