Suppose $\overline{k}$ be an algebraic closure of a field $k$. Usually $G_k \colon= {\mathrm{Gal}}(\overline{k}/k)$ is endowed with the Krull topology. When considering Galois cohomology $H^1({\mathrm{Gal}}(\overline{k}/k), {\overline{k}}^{\times})$, the continuous cochains are taken into account.
Q. Suppose we consider merely a cochain $\rho$ which satisfies $\rho(\sigma \circ \tau) = \rho(\tau)^{\sigma} \rho(\sigma) \in {\overline{k}}^{\times}$ for all $\sigma, \tau \in {\mathrm{Gal}}(\overline{k}/k)$ and is not supposed to be continuous.
Then, is it true that we have still $H^1({\mathrm{Gal}}(\overline{k}/k), {\overline{k}}^{\times}) = 0$?