I looked into this a while back for a mailbox notifier, and concluded that the Energizer Ultimate Lithium batteries had the best low-temperature performance of any widely available type. But my research wasn't really all that thorough. Anyway, I ended up settling on three AAs in series directly powering a Wemos/Lolin D1 Mini Pro. I later gave away that project, but the guy who ended up with it said it survived the big freeze of last winter, and he hasn't changed the batteries. As I remember, those batteries would normally last a number of years. Everything was powered down via a mosfet all the time except for twice a day for a few seconds each, once when the mailman delivered the mail, and once when I retrieved it.
I think the big problem with cold weather is trying to recharge batteries. If you look into Cubesat projects, you'll see they provide for battery heaters. Actually, Cubesat solar power systems are really interesting. You're in bright sunlight for 45 minutes, then in total darkness for 45 minutes. So during the 45 minutes of sun, the panels have to provide enough power to reheat and recharge the batteries, and power the load, and there's some rule of thumb that you can't discharge the batteries at "night" more than 20% or so.
Anyway, if you aren't recharging, it probably doesn't matter all that much which type you use as long as you over-provision a bit on current.