Plants survive the night by producing excess energy during the day. This excess they then convert to starch and store. Research shows that chemical reactions in the plant's cells then calculate, based on their energy store and the rough length of the night, how to proportion their starch stores to last them the night. In the experiment linked above, scientists shut off the light early, giving the plants a shorter day and much longer night (8/16 hrs instead of ~12/12). Still, the plants managed to have enough starch left over in the morning.
Given that plants can adapt to a change in day/night cycle like this, they can easily adapt to a simple double. The day/night proportions remain the same, it's just the time that has changed. Since the daytime has also doubled, the starch the plants store in the day also doubles, so they have twice as much to use at night - meaning they can use the same amount per unit time.
The temperature differences wouldn't have too much effect. A longer night would mean a greater drop in temperature and a greater rise in the morning, but given that we have desert plants that survive +40oC to -20oC, our current plants would either just survive or adapt to the change.