A new gravity independent form of conservation laws for cosmological perturbations is discussed. It is then applied to show that the thermodynamic definition of adiabaticity in general is not a sufficient condition for the conservation of super-horizon curvature perturbations. In slow-roll inflationary models it is a sufficient condition but for other single scalar field models which are adiabatic on any scale (globally adiabatic) perturbations can evolve on super-horizon scales, the Maldacena's consistency condition can be violated and large local shape non -Gaussianity can be produced.
