The volume has to be converted to an implicit surface, or actual geo, pr frame, so the toon edge has a concrete finite border, you cannot detect anything like edges on a regular volume.
Bifrost can load .vdb files --> volume to mesh --> out.
So you can either build the mesh or you can use SDF implicit surface to reconstruct the surface, which the toon edge is able to catch.
Bringing this topic back up because I have a question. Mads suggestion gets you about 40-50% of the way there. You now have geo you can attach a toon shader to, but what does that get you other than a material reacting to a light.
Is there no way you use heat or temperature as the driver for the tonemap. Arnold has the utility nodes to adjust channels when working with volume shaders but is there no way to get a float variable to go beyond that? So that you could ramp yellow at a core, then orange, dark grey, to grey.