That's the thing about divorce... you lose all semblance of control. Neither X has any right nor ability to say how the other should live. As long as they aren't breaking the law, there is absolutely nothing that can be done EXCEPT:
*Sometimes*, and both have to agree, a stipulation in the parenting plan is that new beauxs/women are *either* not introduced or allowed to cohabitate until they've been dating for 6mo. While the horse is out of the barn on this one... it's possible to add it in for any future girlfriends of her exhusband IF he agrees to it. It does go both ways, however.
Seriously, however... he can keep the kids up to midnight every night he has them... or take them on "forbidden" activities (aka forbidden by mum)... or walk around naked all day eating cheetohs. As long as their child is fed, clothed, taken to school, and not being abused, and not committing crimes... neither has any say whatsoever about what happens during the other parent's time.