If you have to push the button three to ten times before anything happens at all, it could simply mean that the battery in the remote is weak -- a condition which can be aggravated by cold weather.
Since it sounds like the garage door actually stops and starts multiple times in order to get the door fully open or fully closed, it means the remote signal is probably okay and the door itself is offering too much resistance.
There are several things that might be happening when the weather gets colder to increase the resistance. It could be the door freezing down, or the springs or door panels freezing together or a problem with the door expanding and contracting with the temperature changes. Either way, when the door itself creates too much resistance, the safety (sensitivity) switch will shut off the motor because the opener thinks the door is jamming against something. The door might not be jammed, but the door controls have no way to determine if that is the case or if something or someone might be trapped by the door so the safety systems will take over.
There is more information and suggestions in the link below that might help further but if the problem remains you might want to consult a professional as the added load can create an unsafe situation and will eventually cause the overworked opener to fail completely.