Anvils do seem to get more brittle when cold, had a friend break the heel off a family anvil once in very cold weather.
Warming an anvil before use in cold weather really helps your forging too, especially things like knives with alloys that have limited forging ranges---getting only *1* hammer blow between re-heats is a pain especially when you factor in decarb during heats!
I know a fellow who simply puts a thrift store clothes iron set on high on his anvil face when he goes into work and by the time he's set up and has the forge hot the anvil is warm to the touch.
Many folks have a "heating slab" of junk steel that they pop in the forge first thing and lay on the anvil when it's red.
The swordmaker I studied under used to take gallon cans with bails on them punch a few holes around the bottom edge and build a kindling fire in them to hang on horn and heel---400# anvil it was a nice warm seat between heats.
As for self I always liked long-johns *NOT* SYNTHETIC FIBER ONES! wool socks and most of all---put down a sheet of scrap plywood to stand on, breaking the cold floor conduction to your feet makes a BIG difference!
A gas forge makes a great shop heater BUT VENTILATE your shop! CO is just as deadly in winter than in summer!