Hey, hey! Ho, ho! That grease has gotta go!
Many years ago, when I was a lad, the NRA published a study on the effectiveness of lubricants in cold weather. Can't find it in my files to refer to. And remember, this was before all of today's super-lubes. But the main ideas still apply I believe.
The gist was that, in any sort of cold weather, you don't want to grease the workings of a firearm. The colder it gets, the truer this is. The NRA article had a graph of the various lubes, and the minimal temp at which each could be depended on to NOT freeze up.
IIRC, paraffin was out from the start. Regular old axle grease was next, and it made the firearm unreliable at something like 40° F, and solidified at a slightly lower temp, to the extent that the firearm wouldn't function at all. Next to go was castor oil. There was some discussion of various SAE engine oils, obviously, the thicker, the higher temp at which they sludged up. Sewing machine oil was discussed as a low-temp lube, but even IT affected firearm performance at a low enough temperature.
Now we're talking well below 0° F, and the only thing they could reccommend was kerosene--yes, as a lube--or powdered graphite. If the powder was to be used in extreme cold, it was reccommended to completely degrease the firearm, and go with the pure graphite.
Personally, and much more recently, I tried Wilson Combat grease in my 1911. Worked GRR-eat! in warm temps, but even on a coolish fall day the slide would periodically fail to close and lock up. Obviously, I de-greased the 1911 and now just oil it, and have never had a failure to completely cycle, since. For all my autoloading pistols I use Outers Gunslick oil, Kroil, Rem-Oil, or S&W spray gun lube, depending on which comes to hand. The Kroil really works its way nicely into the tiniest crannies.
Yr question was on bolt rifles and pump shotguns, and I use the same 4 oils in them. NO GREASE! If I were going hunting in a situation where the temp might get down toward the single digits I'd definitely degrease and go with the Kroil, not one of the others. I've never dealt with shooting in a below-zero F situation, and were that contemplated, more research would be called for. I'm sure the military has something better now than kerosene, for an arctic gun-lube application.