My wife has a 2010 Accent GLS, sometimes when really cold, like 20 or under it wont start, turns over fine, almost sounds like it is out of gas, last time it happened, the check gas cap light kept blinking on and off, ( doesnt always do that though) checked the cap, seemed ok, but still wont start, sometimes go out 20 minutes later and it stats fine. Any ideas ?????

 

If you have a shrader valve on the fuel rail then check for fuel pressure
Check for spark also

 

Could be the O2 sensor. Sounds like a sensor problem. Is it giving you a check engine light? That should throw a code if there is a sensor problem.