  We prove that any faithful Frobenius functor between abelian categories
preserves the Gorenstein projective dimension of objects. Consequently, it
preserves and reflects Gorenstein projective objects. We give conditions on
when a Frobenius functor preserves the stable categories of Gorenstein
projective objects, the singularity categories and the Gorenstein defect
categories, respectively. In the appendix, we give a direct proof of the
following known result: for an abelian category with enough projectives and
injectives, its global Gorenstein projective dimension coincides with its
global Gorenstein injective dimension.
