Let H be a Hopf algebra over a field k with a bijective antipode. We show that the Gorenstein global dimension of H coincides with the Gorenstein projective dimension of the trivial left (or right) H-module k. Then, H is finite dimensional if and only if the Gorenstein projective dimension of k is trivial. We show that, although monoidal Morita-Takeuchi equivalence does not preserve the global dimension of Hopf algebras, it does preserve the Gorenstein global dimension and the Artin–Schelter Gorenstein property. This supports Brown–Goodearl's question of whether every noetherian affine Hopf algebra is AS Gorenstein. Finally, for $H$ and an H-Galois object B, we prove that the module categories $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$ are monoidal model categories regarding Gorenstein projective model structure, provided that the Gorenstein global dimension of H is finite. The corresponding stable categories are tensor triangulated categories. The work is joint with ZHU Ruipeng.