IGH systems use lubricants. Lower budget ones have grease, higher end ones like the Rohloff have oil that should be changed annually. In cold weather, lubricants can get more viscous.

I would not be surprised if in really cold weather that grease could be an issued for ease of shifting, and rolling resistance. I suspect it would have to be REALLY cold before grease causes pawls to get sticky, so I suspect that would not be an issue.

Rohloff suggests in very cold weather that you can use a less viscous mix of oils, but I have never ridden the bike below about 20 degrees (F), and I had no problems at all with using the hub in cold weather with normal (summer) oil.

Decades ago I used a vintage Sturmy Archer three speed hub in winter a lot, no problems. But those vintage hubs had a hole in the hub shell to drip in oil on occasion, thus no viscous grease in the hub.

Cycling cold air to warm air, especially if the warm air is very humid could theoretically allow condensation to form which could cause corrosion, but I do not think that is any worse than riding a bike in occasional rain. That said, Rohloff has warned that condensation of water inside the hub is one reason to perform annual oil changes.

A neighbor is a bike mechanic at a bike shop on a major university campus. He was telling me a few years ago that in spring that year they were replacing an unusually large number of bike chains. I am not sure if that year had more frequent road salt usage or not, but he was saying when a bike came into the shop that needed a new chain, to save time they were using big heavy duty bolt cutters to remove the chains because they were dealing with so many chains that year. That of course is not unique to IGH drive, but it does raise the question of belt drive instead of chain. I am content with chain drive, and I prefer it because for riding around near home I use a 44T chainring, but when touring I use a 36T chainring to obtain a lower gearing range (and adding or removing 4 chain links with a second quick link). If you are going to commute on a bike in winter, I would plan on a chain replacement in spring if you are in an area where road salt is used. I have not done that but there were a few years when I put a worn out chain on my Rohoff bike in fall and discarded it in spring. I am retired, no commuting, only occasional riding in winter.

Shifter cables may have more friction in winter, but that also is not unique to IGH. The higher end IGH units like Rohloff or Pinion bikes use two shifter cables, not one. One cable for upshift and one for downshift. Those systems have the indexing in the hub, thus a sticky or out of adjustment cable is not a problem as that does not impair the indexing. But a single cable hub, the indexing is in the shifter and a sticky cable in winter could be a bigger problem for shifting.

My Rohloff bike was built to use a Rohloff hub, it has an eccentric bottom bracket for chain adjustment. Putting an IGH on a frame that was not designed for it means you need to figure out how you want to do chain adjustment.

I have a sprocket and chainring that both have an even number of teeth. This article explains how half of your teeth wear differently with a single speed or IGH hub.
https://www.sheldonbrown.com/chain-life.html

I am careful to mark my sprocket and chainring to make sure I always put the chain on with the same teeth used for outer chain links or inner chain links. IGH chains can wear much further than the common 0.5 or 0.75 percent elongation point for chain replacement. I do not change chains until well beyond 1 percent elongation. But I suspect less than one out of 10 IGH users bother with marking teeth to make sure that chains go on the same way as before. But if you are a former bike mechanic, you think about that sort of thing.

Good luck.