The mpv process shouldn’t be doing anything while you aren’t doing anything in the app. It sits there idle and quiet for me [Windows, not Linux] until I’m studying a card with audio or adding audio to a note.
Do you have add-ons that are running in the background? If you’re seeing this happen regularly, try running Anki in safe mode – Troubleshooting Checklist#2 – so your add-ons are disabled.
I have the same problem with the official, normal binary release from Github. Perhaps it is not specific to Flatpak.
And uninstalling mpv is perhaps an option sometimes, but not for me as I actually use mpv for actually playing videos. Would be great if there was a way to instruct Anki to ignore/disable mpv integration.
I didn’t test this (and I do not have the problem you’re facing on any of my debian systems) but you could try to use mplayer instead of mpv: Arch Linux - Audio Not Working - #12 by Cassie.