  We show that the Galois cohomology groups of $p$-adic representations of a
direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ can
be computed via the generalization of Herr's complex to multivariable
$(\varphi,\Gamma)$-modules. Using Tate duality and a pairing for multivariable
$(\varphi,\Gamma)$-modules we extend this to analogues of the Iwasawa
cohomology. We show that all $p$-adic representations of a direct power of
$\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ are overconvergent
and, moreover, passing to overconvergent multivariable
$(\varphi,\Gamma)$-modules is an equivalence of categories. Finally, we prove
that the overconvergent Herr complex also computes the Galois cohomology
groups.
