The cold weather itself should cause no problems with or affect your satellite Internet connection.
But when it gets cloudy or snowy then it can block the satellite Internet signal from reaching your satellite internet dish.
Also snow and even ice can sometimes block the satellite internet signal so you might need to clear the ice or snow off your satellite internet dish if it blocks your signal.
Also the weather can sometimes affect the signal to the satellite internet dish through the airwaves.
If you have a strong signal though then the weather shouldn't affect your satellite internet connection as much and if it does then you might have a weak signal and may need your satellite internet dish re pointed.