When using natural carbonation, the fermentation process restarts, although in a much smaller scale. The yeast must ferment the sugar, then clean up the byproducts of fermentation as in the secondary phase. Because the yeast population is much smaller, the process can take up to and beyond two weeks for full carbonation. Once the bottle/keg conditioning phase ends, the beer begins aging. Aging typically rounds out any rough edges in the beer and can remedy many imperfections.
It would then logically follow that your (bottled) temps should have some kind of a connection with the fermenting temps.