Frozen cables: I've been there. Riding off-road in 20-degree F weather, I once tried riding over a narrow rivulet on a sheet of ice that, it turned out, wasn't quite thick enough. Luckily, it wasn't too deep, so I got soaked only up to my knees. The bike was completely submerged, though. The cables froze instantly: the gears didn't work, the brakes didn't work. I had to bash the front brake cable housing between two rocks to get that to work at least. Again luckily, the car was only a 15-minute ride away, and I'd been riding hard and was very warm, so no ill effects. It was interesting to watch steam coming off my feet and legs as I rode back to the parking lot.
Here's what I'd do with your bike:
After noting how the inner wire is routed at the rear derailleur pinch bolt, pull the inner wire out of the rear derailleur cable housing (leaving the housing in place). Confirm that neither the housing nor the inner wire has kinks or frays that might add friction.
Might be best to simply install a new inner wire or new cable housing or both at this point, as cheap insurance against future failure. (If you do decide to replace the housing, leave the inner wire in place temporarily and slide the housing out of the frame. That way, you can use the inner wire to guide the housing back through the frame.)
Drip some light oil (Triflow or equivalent) into the cable housing at the shifter end. Reinsert the inner wire and run it back and forth in the housing a few times to distribute the lubricant evenly in the housing. Reconnect the cable to the rear derailleur and adjust the tension to ensure correct indexing.