  Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for
any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and
only if the underlying left $R$-module $_{R}M$ is Gorenstein projective
(injective). (2) if $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M<\infty$, then
$\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M =
\mathrm{G}\text{-}\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective
dimension also holds. (3) if the extension is split, then
$\mathrm{G}\text{-}\mathrm{gldim}(A)= \mathrm{G}\text{-}\mathrm{gldim}(R)$.
