Ideally you would not change the computer at all so your type certificate doesn't change. If you have to (or for commercial reasons really want to) make a change you would try very hard to keep that the same type certificate or at most a variant of the same type certificate. If you can do that then it will be flown with the same type rating and you avoid all the crew training cost issues.

But to do that you'll still have to prove that the changes don't change any of the aircraft characteristics. And that's not just the normal handling but also any failure modes. Which is an expensive thing to do, so Airbus would normally not do this unless there is a strong reason to do it.

The crew is also trained on a lot of knowledge about the systems behind the interface, so they can figure out what might be wrong in case of problems. That doesn't include the software architecture itself but it does include a lot of information on how redundancy between the systems work and what happens in case one system output is invalid. For example how the fail over logic works in case of a flight control computer failure, or how it responds to loosing certain inputs. And how that affects automation capabilities, like: no autoland when X fails, no autopilot and degradation to alternate contol law when Y fails, further degradation if X and Z fail at the same time. Sometimes also per "side", not all computers are connected to all sensors.

The computer change can't change any of that without requiring retraining.