I would test for ammonia, nitrite, and nitrates. One thing many forget about fish keeping is that these animals grow, and what was fine for an environment when they were young doesn't always remain so. That is still a lot of koi in a pond of that size, and as they get larger, so does the output of waste. In an outdoor environment there are other factors that play into the waste issue... leaves and other debris that get into the pond can raise waste levels quickly. The weather can contribute... washing other forms of waste into the water, too.
As for the water temp... 5 degrees is a lot for a fish to handle if it's a quick change. Without being able to examine one of the fish, or perform an autopsy, diagnosis will have to come in other forms. How I wish I could see them. An autopsy can tell so much.
I know this is a horrible thought, but could someone have put something into the water? I have seen cases of poisoning, intentional... where the fish have died for no apparent reason, and we weren't able to determine cause of death until we performed the autopsy. In the instance I mention, someone had poured a clear chemical into the water, I forget which chemical... but overnight, perfectly healthy 12 - 14 inch koi suddenly dropped dead. This woman lost 10 fish.
Other than poisoning, water quality is the only other thing I can think of. Temp WILL have something else to do with it, also.... As the fish get larger, they consume more oxygen. Oxygen depletion will cause the symptoms you have described as much as will waste issues or poisoning.