For the Gorenstein homological properties under changes of rings the Gorenstein cotorsion property of modules and the preserving property of the corresponding dimension under a separable Frobenius extension of the ring were proposed. It was first proved that for a separable Frobenius extension R→S the S-module M was a Gorenstein cotorsion module if and only if M was an R-module of a Gorenstein cotorsion module and thus the Gorenstein cotorsion dimension of modules remain invariant along such ring extension. As an application it was shown that if this ring extension was splittable the overall Gorenstein cotorsion dimension of the ring remained invariant as well. In addition the Gorenstein cotorsion dimensions over group rings were discussed and the invariance of the Gorenstein cotorsion dimensions under the separable Frobenius extensions was further verified