  Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we first
introduce the $\xi$-Gorenstein cohomology in terms of
$\xi$-$\mathcal{G}$projective resolutions and $\xi$-$\mathcal{G}$injective
coresolutions, respectively, and then we get the balance of $\xi$-Gorenstein
cohomology. Moreover, we study the interplay among $\xi$-cohomology,
$\xi$-Gorenstein cohomology and $\xi$-complete cohomology, and obtain the
Avramov-Martsinkovsky type exact sequences in this setting.
