Koi need to be in water above 74 F for thier immune systems to actually work properly. Medicines and treatments for any problems do not work at 65 F. Digestion completely stops at 45-50 F. If they eat and have food in thier digestive system it will rot and they will die. Koi thrive in water temps above 75 F.
Trout thrive in cold water. The colder the better, down to 40 F. When they are thriving, koi cannot do anything but lay there dormant. When Koi Carp are thriving, Trout are floating belly up, dead. This is not a it depends subject.
What medicines or other treatments would you expect to be delivering to KOI that are in a large trout pond? I'd expect that would never be an issue?
You'd have a bigger problem of the turbidity of KOI waters affecting the trout negatively then you would with temperatures as long as you stay in a safe range for the type of trout. Koi can survive an incredible temperature range so the odds of killing them off with temp variations that trout would survive would be limited at best.
I dunno TPF but I'd say everything is an 'it depends subject'.
If the OP just wants both species to survive and can provide a middling temp with enough D.O. then it is entirely doable. Optimum for both, not likely, but that is the case in almost all bodies of water/species combos. Of course the thermocline in a pond will in theory allow the fish to sit at their most desired available temp.
