  We prove a large deviation principle for the slow-fast rough differential
equations under the controlled rough path framework. The driver rough paths are
lifted from the mixed fractional Brownian motion with Hurst parameter $H\in
(1/3,1/2)$. Our approach is based on the continuity of the solution mapping and
the variational framework for mixed fractional Brownian motion. By utilizing
the variational representation, our problem is transformed into a qualitative
property of the controlled system. In particular, the fast rough differential
equation coincides with It\^o SDE almost surely, which possesses a unique
invariant probability measure with frozen slow component. We then demonstrate
the weak convergence of the controlled slow component by averaging with respect
to the invariant measure of the fast equation and exploiting the continuity of
the solution mapping.
