My 2004 Allroad 4.2 V8 Audi

Fuel pump or Crank Position sensor. A failing CPS will have issues during a warm start after its been heat soaked. I’d try swapping this first with a quality part from FCP Euro A failing pump with a bum check valve may also cause this condition. A weak / old pump might not be able to regenerate adequate fuel rail pressure once it’s hot (and the check valve that’s bad has let it bleed back into the tank). From an electronics perspective, heat = resistance. So a weak pump might have issues once it’s warmed up