We could have it like singleplayer levels, showing best times for each
Don't think this makes sense for an Idle game. It's not like SP where you have # of moves as a measure. Amount of time for levels that take multiple days and have sleeping/eating in between seems like it's simply giving recognition to those who (A) have the highest AP [in which case just make that the measure], and (B) play the most actively.
So once someone who has Ascended 3, 4, 5 times takes another pass through the levels, no one who also hasn't had multiple ascensions will be able to beat the time. Effectively eliminating 95%+ of players from even being contenders.
Compared to the SP, where # of turns means anyone who figures out a better way of doing it can still be a contender. You don't stronger on your 2nd playthrough of the SP campaign, able to improve your previous results.