Most issues with cold weather opening are actually related to the door itself. Often when temperatures go down the bottom edge of the door freezes to the floor or weather stripping or the door contracts until it binds in the lift mechanisms and in certain conditions the joints between the door panels can even freeze together. All of that adds to the load the opener has to lift and it can become too much for the opener to overcome.
Next time the temperature drops enough for the problem to occur, try releasing the door from the opener with the release rope and see if the opener will move the trolley back and forth by itself. If so, then the door likely needs some adjusting or attention. The balance of the door may be off during cold weather or the lift springs themselves may be at fault. There could also be a problem with the door-closed position if the door or floor or any weather stripping contract or shift at that temperature and create binding on the door when in the closed position.
If the door balance and weight test good and the door moves easily by hand, then another possibility is excess drag in the opener itself due to a failing component. Or a damaged logic board or other electrical connection could be failing when cold. Some models have a fault code LED that might flash on the opener itself to indicate a problem area but it can be difficult to test the components when the problem occurs only at certain temperatures, so you might consider consulting a professional in that case.
There may also be a combined problem of some added door or opener drag due to the cold and a slight misadjustment of the force settings on your opener. Different models have different methods for adjusting those settings but even relatively small openers and doors can create enough force to do a lot of damage or injure someone if the settings are off, so again you may want to consult a professional at that point.
The links below have additional information that might be helpful.