Yes cold weather does have an affect on the TPMS system of your vehicle.
During cold weather the air in your tires can be come more dense and which can make your tires seem low and then trigger the TPMS light to come on thinking your tire pressure is low.
So when you drive the vehicle for a mile or two then the tires will start to warm up and the air will usually become less dense which will allow the TPMS system to realize there is enough air pressure in the tires.
Then after a few miles of driving the TPMS system light should turn off and if not then something is wrong with the TPMS system if your tires have enough air.