  In the context of single-field inflation, the conservation of the curvature
perturbation on comoving slices, $\R_c$, on super-horizon scales is one of the
assumptions necessary to derive the consistency condition between the squeezed
limit of the bispectrum and the spectrum of the primordial curvature
perturbation. However, the conservation of $\R_c$ holds only after the
perturbation has reached the adiabatic limit where the constant mode of $\R_c$
dominates over the other (usually decaying) mode. In this case, the
non-adiabatic pressure perturbation defined in the thermodynamic sense, $\delta
P_{nad}\equiv\delta P-c_w^2\delta\rho$ where $c_w^2=\dot P/\dot\rho$, usually
becomes also negligible on superhorizon scales. Therefore one might think that
the adiabatic limit is the same as thermodynamic adiabaticity. This is in fact
not true. In other words, thermodynamic adiabaticity is not a sufficient
condition for the conservation of $\R_c$ on super-horizon scales. In this
paper, we consider models that satisfy $\delta P_{nad}=0$ on all scales, which
we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where
$c_s$ is the phase velocity of the propagation of the perturbation. A known
example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$.
In order to generalize USR we develop a method to find the Lagrangian of GA
K-inflation models from the behavior of background quantities as functions of
the scale factor. Applying this method we show that there indeed exists a wide
class of GA models with $c_w^2=c_s^2$, which allows $\R_c$ to grow on
superhorizon scales, and hence violates the non-Gaussianity consistency
condition.
