j9kmo7ed Aurora Store can be obtained from their Gitlab repository: https://gitlab.com/AuroraOSS/AuroraStore/-/releases
Yes, I've seen that api message appear for many apps I've installed. Are there levels to how much "integrity" an app requires, since these apps accepted my phone?
There is information on that here (under the "Not compatible" headline): https://discuss.grapheneos.org/d/8330-app-compatibility-with-grapheneos#:~:text=issuecomment%2D1766887298%29-,Not%20compatible,-Due%20to%20the
An app might not have enabled the Play Integrity feature for their Play Store entry in the past, but it sounds like Air Europa now has.
I'm not sure I'm the best at explaining Play Integrity, but I'll make an attempt.
Basically, the Play Integrity API feature in Play Store means that the app blocks installs from the Play Store when the Play Store detects that the OS that you're using doesn't license Google Mobile Services. In simpler terms, if the OS you're using doesn't give Play services the high level of system access it has on, for instance, the stock Pixel OS or the stock Samsung OS, it won't pass various checks which the Play Integrity API requires it to pass.
Note that there's a difference between app developers enabling Play Integrity for their app entries in Play Store, and enabling it for actual usage of their app. Vipps and BankID, for instance, don't use Play Integrity for their Play Store entries, but use it when you open or interact with their apps in certain situations (but those apps don't enforce Play Integrity on the strongest level, which GrapheneOS doesn't pass).
There was a recent change to the Play Integrity API which adds two additional checks in order for the user to be able to pass the most basic integrity check (called "MEETS_BASIC_INTEGRITY"): You now have to install those apps from the Play Store, and you need to be signed in to the Play Store. Otherwise the basic check will report a fail. This is why Vipps started reporting – for some users – that their devices are rooted.
You can test this for yourself with this app: https://play.google.com/store/apps/details?id=gr.nikolasspyr.integritycheck
If installed from Play Store, that app will report that your device passes the basic check; if you install the app from Aurora, the check will fail.