  We construct cohomology theories for $(\varphi, \tau)$-modules, and study
their relation with cohomology of $(\varphi, \Gamma)$-modules, as well as
Galois cohomology. Our method is axiomatic, and can treat the \'etale case, the
overconvergent case, and the rigid-overconvergent case simultaneously. We use
recent advances in locally analytic cohomology as a key ingredient.
