Comment 10 for bug 1970282

For a Flatpak application to follow the system theme in use, the corresponding Flatpak theme needs to be installed. For example, if I use the "Yaru" theme in "Dark mode" with the "Prussian Green" accent color, the "org.gtk.Gtk3theme.Yaru-prussiangreen-dark" needs to be installed. If it is, both system applications and Flatpak applications look consistent.
However, only a few themes are available in the Flathub repos. If you search for Yaru themes on Flathub (flatpak search Yaru), you will notice there are alot of variations, but only the packages not presenting a "Version" number correspond to the new 22.04 Yaru theme. As such, only the following Yaru (22.04) themes have a correspondent Flatpak theme:

Yaru
Yaru-dark
Yaru-sage
Yaru-bark
Yaru-bark-dark
Yaru-prussiangreen
Yaru-prussiangreen-dark
Yaru-olive
Yaru-olive-dark
Yaru-magenta
Yaru-magenta-dark

For a consistent look between Flatpak and non-Flatpak applications in Ubuntu 22.04, the missing accent color themes need to be available in Flathub.