No way to prevent automatic lookup of Flatpak updates
Perhaps this is by design, but there appears to be no way to prevent Software from automatically searching for updates to Flatpak apps. Even when turning off Automatic Updates and Automatic Update Notifications, Software will still search for Flatpak updates in the background and send a notification if any are available. Though I suppose those toggles are strictly for preventing installation of updates, not discovery of updates.
Perhaps there should be an extra setting to automatically check for updates, to go along with the existing Automatic Update setting. If that would be too many settings, then Automatic Updates could instead control whether automatic update searching is performed. This behaviour could cover package-based apps as well.