  We study the long-time behavior and the regularity of pathwise entropy
solutions to stochastic scalar conservation laws with random in time spatially
homogeneous fluxes and periodic initial data. We prove that the solutions
converge to their spatial average, which is the unique invariant measure, and
provide a rate of convergence, the latter being new even in the deterministic
case for dimensions higher than two. The main tool is a new regularization
result in the spirit of averaging lemmata for scalar conservation laws, which,
in particular, implies a regularization by noise-type result for pathwise
quasi-solutions.
