Following calculations are for Dave's scenario where community cards from the first run are not used in the second run.
The odds AKo vs QQ are 43 : 57
In one play 43% of the time AKo gets all the money and QQ gets nothing, 57% of the time the other way around.
During the first run the most common situations are:
30% A/K hit and Q didn't -> 39 : 61 in the second round
4.8% AK hit and Q didn't -> 33.5 : 66.5
3.2% AA/KK hit and Q didn't -> 33.5 : 66.5
---------------
38% noone hit -> 44 : 56
10% Q hit and A/K didn't -> 47 : 53
6% Q hit and A/K hit -> 42 : 58
Other outs are 1% or less, so I didn't take them in account because it's too much work
.
Due to QQ needing far less outs to win it's chances in the second round don't diminish much when it wins the first round, odds to win the second round are 44 : 56.
When AK wins the first round it will have used a bunch of its outs, so it can't use those outs in the second round, which means odds are reduced to 38 : 62.
So running twice for a $100 pot will result in:
16.3% win/win for AKo
26.7% win/lose for AKo
25.1% lose/win for AKo
31.9% lose/lose for AKo
Total: 42.2$ for AKo and 57.8$ for QQ (vs $43 : $57 in a single run)
So running twice and not reusing the cards is bad for whatever
hands needs most outs to win.
In tournaments you might want to run twice anyway when you have the smaller stack and are all in, because running twice drastically cuts the chances of you getting kicked out of the tournament.