I've read of a lot of different barrel break-in procedures, and none of them have ever even discussed this subject. I believe the answer is no, or if there is an ideal temperature, the answer is - it remains unknown.
Not sure, but am guessing your cleaning products will work more effectively at warmer temps. I think a big part of breaking in a barrel is knowing how to clean a barrel effectively and then actually effectivly doing so between shots or groups of shots, so that may be something to keep in mind. Your cleaning products will probably be quicker and more effective at room temps vs. 20F, for instance. Just my .02.
I dont think the outside air temp has much to do with breaking in a barrel. I think letting it cool between shots and getting it good and clean during the break in period has more influence than what the thermometer reads. I have found when I break in a barrel that by the time I get it clean it has cooled off pretty well.
Break the barrel in when the temperature is most comfortable for you...the barrel and cleaning products won't care. For me when it's too too cold my fingers don't work to well with all that small stuff-patches, brushes and when it's too hot well I sweat all over my cleaning stuff.
Thanks so much for the info, the weather has been in the high 30's to low 40's and I have been waiting for it warm up. I hadn't thought about the solvents, but that is great insight that comes with experience.
When everyone talk's about letting the barrel cool off between shot's, I would imagine that barrel thickness and outside air temperature will play a role in how soon the barrel will cool and allow you to fire a second or third shot.
I've broken in barrels in -20F. I step outside the door, shoot into a stump in the back yard, step back in the house and strip the bore down to bare steel. If you can shoot in your yard, you can break a barrel in regardless of outdoor temperature - extreme hot or cold. My barrel is more or less room temperature no matter the extreme variation in outdoor temps because the barrel and cleaning solvent are in a temperature controlled setting 99% of the time. Of course you may not be able to shoot at your residence, then you do have to consider creature comfort, or look for a relative or buddy located where you can set up and shoot 10 times without being reported to the local Police Department.