Please note: I am NOT part of the GrapheneOS project in any way, I'm not working with them in any way, and I don't represent their views in any way. I am just a user contributing my knowledge. For transparency sake, every time I've previously contributed my knowledge about the Auditor app/attestation on this forum, I got my posts deleted by moderators who believed that I'm spreading misinformation, which I believe is incorrect and unjustified. What I'm posting here is what I believe to be correct, but feel free to wait for others to weigh in. I think the Auditor app is great to have, it also allows existing GrapheneOS installations to verify other GrapheneOS installations without resorting to manually verifying the key from the website (which relies on the HTTPS/root CAs security model which is problematic) so it's important to have it preinstalled in GrapheneOS, and they recently extended the Auditor app to report some security-sensitive settings, which is cool. I still think the Auditor app could be described better to users to prevent people from falsely believing that the Auditor app can verify the volatile running copy of the OS in RAM. See below for details.
You say you verified the verified boot key fingerprint displayed at boot. That's great! My recommendation is simple: Set up the Auditor app, check that the security patch levels are up-to-date, and if you suspect anything malicious has occurred in privileged OS components, simply reboot and that's all. After installing GrapheneOS updates that include a new monthly security patch level, check the attestation result and ensure that it reports the updated patch level as well.
axino I thought verifying the boot key hash from the screen was enough.
It largely is. Once you install the correct GrapheneOS verified boot public key and lock the bootloader, there shouldn't be any way to change this key without triggering a full wipe, short of compromising the secure element.
Depending on your threat model, you might have preferred to set up the Auditor app as soon as possible. You can boot into Safe Mode which prevents the OS from loading third-party apps until the next reboot before any third-party app can start, and also automatically enables Airplane Mode, hopefully before any network communication is done (it should be enabled before network communication, but the devs have said in another thread that this may not be guaranteed due to the baseband firmware), which should prevent any external attacker or locally installed exploit from compromising the OS using an unfixed vulnerability after the verified boot process checks the OS during boot. But again, depending on your threat model, it might not satisfy you much, if you think an attacker has already compromised the verified boot process or can exploit the OS even though all third-party apps should be temporarily disabled, etc. This should only be necessary for the initial pairing, because after that the Auditor app will ensure that the attestation results are coming from the specific secure element in your specific device that it will now cryptographically recognize in all future attestations (the only way to bypass this would be to compromise the secure element, but then you couldn't trust neither the verified boot process nor any attestation results at all).
axino Auditor Attestation will periodically check (every 6h for remote with 32h possible delay in notification) if there was corruption or difference.
Remote attestation (with the Auditor app or any app performing remote attestation really) doesn't and cannot check the volatile running copy of the OS in RAM. The secure element can attest the properties it has like the verified boot key installed into it, and can attest things that are part of the verified boot process like the OS's security patch level, as part of the "hardware-verified" information. Once the OS is booted, the secure element can gather information collected by the OS into a separate extension bundled into the final attested response from the secure element, such as sensitive configuration like the USB-C port and OEM Unlocking setting from the OS, but the Auditor app rightfully separates it into a separate "OS-verified" section because this information is gathered from the running OS, so the secure element relies on the OS to tell it the truth and cannot itself attest it. This is useful to prevent malware that compromises or spoofs the display on the device, for example a website that mimics your device's display that you accidentally entered into full-screen mode and you don't notice is still in full-screen mode, from telling you fake information, because you know that the secure element gathered it directly from the OS that was already verified by the verified boot process, but it still has to trust the OS for it and if you suspect that the OS is compromised post-boot using an unfixed vulnerability in a privileged OS component, then this might not be useful to you.
The secure element cannot reverify the volatile running copy of the OS in RAM. The verified boot process verifies the stored copy of the OS before it's loaded to RAM and starts executing. This is because the verified boot process relies on a chain of trust where the integrity of every component is checked by the preceding component, tracing all the way back to the boot ROM that's burnt into the hardware of your device at factory. Because of this, once execution is transferred to the next component, it cannot be checked again short of rebooting the device to restart the verified boot process from the hardware root of trust.