The starting failure may be due to multiple things.
In the first place you have to make sure that the car continues to have a hot spark when it turns hot with the starter engine, to remove a spark plug and give candle to the key is quickly seen.
Discarded the electric theme, you can try to put a drizzle of gasoline by the gas butterfly when you give you the failure, if the engine starts, and stop, the problem is definitely injection.
And now the mess. If you say that it starts well and hot, it can not be as simple as the circuit does not maintain the injection pressure when you stop the car, because the anti -background valve of the pump, the pressure regulator that goes after it and before the filter, or the vacuum pressure regulator that is in the injection ramp next to the injectors are bad and do not close tightly.
In this way, when the car stops in normal conditions, gasoline should be at a pressure of about 2.5 to 3 bars in the tubes. In your case, the tubes stay at 0 bars, but full of gasoline.
This makes you start with the cold engine, gasoline is also cold and liquid, so that the pump immediately press the system and the failure goes unnoticed.
But when you start hot, if gasoline is not under pressure, the heat of the engine causes it to boil inside the tubes, causing them to empty completely. When you start the gasoline pump, the circuit has to load until the gasoline reaches the injectors.
To check this, just let the engine heat, stop it, leave it for a while and where it should fail when you start again, put the contact three or four times in a row without starting the car, (the characteristic buzz of the gasoline pump will be heard for a couple of seconds) in this way the circuit again of gasoline will be loaded.
Already then turn to start, if the engine starts, the failure is that.
Another thing that can also happen is that all the components are well and have some chopped tube, but it would smell gasoline and have it in progress would drip gasoline to the ground.
I hope it helps you.