Hello,

This is a classic Audi V8 problem.

The crankshaft position sensor and/or the engine speed sensor is faulty.
They are located on the rear of the engine block by the flywheel beneath the left exhaust manifold.

They are permanent magnets and over time they deteriorate because of the heat from the exhaust manifold.

To test them you measure the resistance.The connectors have 3 terminals. Between no 1 and 2 resistance should be approximately 1000 Ohms. Between 1 and 3 infinite and between 2 and 3 infinite.

The crankshaft position sensor is the one causing the difficulty to start a warm engine but propably the other one is also gone.

The cost is - in Sweden - 200 € each.

If you can read german - look for "OT-Geber" in the forum and in the FAQ.

Good Luck!
Regards
Anders