As far as being cold goes there's not a battery in the world that I know of that isn't affected, lithium, alcaline, nicad,lead-acid you name it. You only have to listen to cars starting up on a good frosty morning to realise what effect the cold has.
If it's possible (with out undue wear and tear on the battery cover) pop the battery out when not in use, and before replacing it in your vernier warm it up in your hand you should be able to extend battery life. This also stops corrosion often found in torches and the like.
Some verniers and other measuring devices have a solar array which helps as long as they are in a strong light source. In which case keep it under a light sourse as long and often as possible (without it being on). Occasionally Model engineer magazine give away a digital vernier but you do have to take out subscription for a year and ofcourse it is a one and only one time offer.
When replacing batteries try to obtain them from someone who has a large turnover in batteries, ensuring they are fairly new. Don't buy lots of batteries at once only enough for your needs. (refer previous note on supplier) Together these little tips may help keep your verniers going.
The best compromise I have found is a digital readout (numbers) which has mechanical movement like an odometer but that is only on a micrometer of mine. As backup I have a set of normal verniers but I also need my bifocals on to read them accurately these days.
Jesse.