Starts and runs perfectly when cold or hot. When thermostat starts to open and temp goes through a certain temp range engine sputters and may back-fire. Then runs great. Throws a 420 code which is fuel-air ratio, O2 sensor, bad cat converter. If you shut it off at this in between temp or it cools down to this in between temp it will not start and you smell gas as if it is still trying to cold start therefore pumping in too much fuel even though it is somewhat warm. If you let it cool down completely it starts and runs fine again. If you re-start it hot it also starts and runs fine.
New fuel pump, new cat converter, new O2 sensor, new engine temp sensor, new plugs, new distributor cap and rotor, new thermostat. What next? It's only at that in between temp range.