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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 $\zeta$.

Adiabaticity and gravity theory independent conservation laws for cosmological perturbations