of course there could be many problems. but "heat soak" is one, IME, that fits the bill here as well as any other.
"heat soak" is what happens after an engine is fully warmed up, driven for a while, and then the key is turned off. when the ignition is turned off and the with the hood closed the temperature inside the engine compartment rises rapidly within a few minutes from the radiating residual heat in the heads. it should not be surprising, after all, the water pump is not turning and so there is no coolant moving, the fan is not turning and there is no ambient air blowing through the engine bay. all three, in combination, are normally what keeps the engine from massively overheating. this is rarely noticed as most people leave their cars after the turn off the ignition.
as the temp in the engine compartment rises, so does the temperature of the fuel in the floatbowl and at some point it will become so hot that it will be partially atomized even before it is mixed with air. if one attempts to start the car at that time, the fuel/air mixture will be too LEAN to combust properly. and "that time' is actually an interval that may start less than ten minutes after engine shutdown and extend for up to an hour.
and oddly enough, enrichening the mixture a bit with the choke can actually permit the car to start. but it may run very rough for up to thirty seconds or so before a cooler mix of fuel reaches the jets.
the only remedy i've found if one suspects the conditions are ripe for "heat soak" is to develop a habit leaving the hood open a foot or so after turning off the ignition until one returns. there are other remedies such as installing an auxiliary fan or something similar.
this is usually a problem in carbureted cars only. and usually only a starting issue. once started if the car continues to run poorly, it's most likely due to something else. and if opening the hood fully doesn't fix the problem, again, it's most likely something else.
