As another thread regarding this theme was deleted, here is the solution for anonymous accounts in Aurora:
In Aurora go to "Settings → Network → Dispenser management" and delete the dispenser URL, but save the URL for possible future use.
In GrapheneOS go to "Settings → Apps → See all apps → Aurora Store", click on "Force stop", go to "Storage and cache", click on "Clear cache".
Then open Aurora and it should work again, but don't logout or request a new session, it will fail because of the deleted dispenser. If this accidentally happens, re-enter the previously dispenser URL in Aurora settings, refresh the account/session and repeat the steps mentioned above, if necessary.