This seems like an odd question but in the past I've always made compost in a bin and left it there as I used it. At the moment I have some Berkley method compost that was recently finished. I've left the cover on it as I don't need it just yet, but am wondering how best to store it. Do I treat it like an active pile (keep it moist and warm)? It's very hot, windy and dry here and it's in full sun.
I'd just keep it covered, perhaps two or three layers of covering materials.
I've kept finished compost in a heap with old carpet for a cover, works quite well.
You can use a first layer of cardboard for more retention, the cover just needs to be able to hold moisture but also breathe.