On my car it wasn't exclusively hot - it could start-then-die either hot or cold.
I sent the car to a main dealer and asked them to check for faults (summer '14), but nothing was found.
Then subsequently (later that summer) it went back to main dealer for some other work.
At that time I asked for any software updates to be done.
One of these was a gearbox update.
Since that time, the problem has never re-appeared.
I couldn't say which updates may have cured it, if that is what did happen ...
But consider trying this:
If you press 'start' and then immediately pull the auto gearshift back, the engine will stop.
In other words, by doing this you are replicating what happens when the car starts-then-dies.
The systems are telling the car not to start (because the gearlever is not in 'park') just at the time when the engine has fired, but is not yet properly running.
I'm thinking: this would be enough to cause software to stop the engine.
I'm thinking: perhaps the programming of the software update addressed this.
If I were you, I should check if there are gearbox updates that could be applied to your car.
Or, belt and braces, any updates at all. The quick brown fox

My car is also now fixed (last 3 months) in my case it seems to be a transmission module software update that fixed it. Reading other posts it looks like this may be the culprit - telling the ecu that the gearbox is NOT in park and therefore will not allow a start.

Good luck if you still have this issue- I am convinced it is now finally sorted!!!

Paul Current - MY 2007 HSE i6 3.2 Auto, Baltic Blue
Previous - MY2005 Discovery 3 TDV6 S, Bonatti Grey
Also own - 2011 Jaguar XF S 3.0D Polaris White

Yep 20 months plus and not a problem since I6 WITH GAS CONVERSION
HST BODY KIT
OSRAM LED DRIVING FOGS
CARS LOOKING GOOOOD!