  We investigate the stationary measure $\pi$ of SDEs driven by additive
fractional noise with any Hurst parameter and establish that $\pi$ admits a
smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The
proofs are based on a novel representation of the stationary density in terms
of a Wiener-Liouville bridge, which proves to be of independent interest: We
show that it also allows to obtain Gaussian bounds on the non-stationary
density, which extend previously known results in the additive setting. In
addition, we study a parameter-dependent version of the SDE and prove
smoothness of the stationary density, jointly in the parameter and the spatial
coordinate. With this we revisit the fractional averaging principle of Li and
Sieber [Ann. Appl. Probab. 32(5) (2022)] and remove an ad-hoc assumption on the
limiting coefficients. Avoiding any use of Malliavin calculus in our arguments,
we can prove our results under minimal regularity requirements.
