Depends entirely on the aircraft type, and the operator.
On my type, we use the APU to start both engines. Provided that we dont need it for pressurisation on departure we shut it down as part of the after start checks. If we use it during the departure, we switch to ENG BLEED and shut it down as part of the after take-off checks.
We use it for departure in situations where we need all the engine performance we can get, and it will happilly run the pressuriastion but not anti ice.