Will not stay running

More likely a crankshaft position sensor or camshaft position sensor is failing. The glaring clue is the temperature variable. Those sensors commonly fail by becoming heat-sensitive, then they work again after they cool down for about an hour.

The engine computer looks at those two signals to know when the engine is rotating, then it turns on the fuel pump, injectors, and ignition coils through the automatic shutdown, (ASD) relay. When one of the signals goes missing, the computer turns those systems off. To add to the misery, it can set a diagnostic fault code related to each of those sensor circuits, but they often do not set in the little time it takes for the stalled engine to coast to a stop. The way to approach this then is to connect a scanner so you can view live data and see what the engine computer is seeing. If a sensor's signal stops showing up, you will see the reported state change. On my Chrysler DRB3, that changes from "Present" to "No". It will continue to show "No" during cranking.