I only blanket when it gets colder than -10 C. I was told by my farrier/equine nutritionist that horses' metabolisms do not have to work any extra unless it gets colder than that. I suppose it would depend on what the horse is used to, to some extent, but I would certainly not blanket unless it was below freezing. The idea of putting a blanket on in the late afternoon just in case the temperature drops a few degrees at night is problematic. The last thing you want to do is for the horse to sweat from the heat of the blanket and THEN get cold. Better to let the horse adapt on its own. He will be fine and will not grow a winter coat at 13 C! That is summer weather to us! A difference of 10 degrees, when the temps are staying that warm, will not cause him to get cold. If anything, he's probably MUCH more comfortable at 13 without a blanket, than at 25!
Other than overheating, blankets are problematic for other reasons. They cause rubs, can get caught on things, can create bacteria from reduced air flow. Leave your horse unblanketed would be my advice!
P.S. I also have dutch doors exactly how you describe. I shut my horses in, because tonight, for example, it will be -17. But in the summer, they come and go as they please. He will NOT be cold in 13 C weather inside or out, without a blanket!