As already requested, please tell us your oil level.
When the low oil pressure warning comes on, damage is being done to your engine, so the engine needs to be stopped immediately. Since it happened twice, that’s not good. If the issue isn’t a very low oil level, you should have this car towed to your mechanic to prevent further engine damage.
You state engine was off, was the key in the engine on position? When key is turned to ‘run’ without starting the engine ALL warning lights should illuminate.
Or, did the engine stop when the oil light came on?
Please expand on your question with more information.
As far as the portion of your question about the outside temperature, my response—unlikely.
If the engine was not low on oil, and never run when low on oil, the OP sending unit is a likely failure point. These are usually near the oil filter, and fairly inexpensive and easy to replace. Good luck and please let us know.