I suspect when it comes to the evolution of the universe we are starting to run up against the edge of the reference class that the Copernicaln principle acts within and start seeing anthropic effects. See here - star formation rates are falling rapidly across the universe and if big complicated biospheres only appear within a few gigayears of the formation of a star or not at all, then we expect to find ourselves near the beginning despite the universe being apparently open-ended. This would have the side-effect of us appearing during the 'priviliged' era.