When you turn on the ignition do you hear the fuel pump run for a few seconds ? I guess that you already know that the pump runs for a few seconds to pressurise the fuel header, and then the ECU turns it off, once the engine is turning over, after a short delay to allow oil pressure to build up, the ECU turns the pump on. A non running fuel pump could be the inertia switch, fuel pump relay, fuse, or a faulty fuel pump.
On my 3.9 I have had the following problems that tended to be affected by under bonnet temperature
Rotor arm - gave intermittent starting problems and eventually a complete failure to start
Pickup in the distributor - again intermittent starting issues - eventually cured by passing a piece of cardboard through the gap to clean up the pickup and the toothed wheel, I did then replace the pickup
A non temperature related problem was the connector between Morgan wiring loom and engine (Land Rover) loom. Simply unplugging and plugging back together fixed the problem. A dose of contact cleaner was then applied to 'improve' the fix
Another temperature related issue others have experienced is a faulty ignition coil that gets faulty when hot
We have lost the sun today, hoping for some rain, our garden needs it
