Wow...they did a good job. I like that they used white EPDM. They
weren't sure what the gray stuff was -- it was probably PVC. TPO is the
latest common roofing membrane for buildings.

They didn't glue it properly, unless they used some glue with which I'm
not familiar. I'm the roofing business, and we do some EPDM roofs but
not a lot. The proper way to use common solvent-based or water-based
glues is as contact cement -- you spread it on both surfaces, wait for
it to dry, and then stick it and hope you got it right the first time
-- because if you didn't, it will have to be good enough.

On buildings, we don't glue it to wood. It gets glued to insulation.

The insulation, in residential applications, is often 1/2" MDF (Medium
Density Fiberboard) which is light weight, sized in 4x8 sheets, and
very dusty with this awful fiber dust that sticks to your sweat (and no
matter how cold it is, I sweat just looking at this crap and thinking
about it sticking to me) and makes you itch.

In bigger, higher-budget applications (we do some government work, but
rarely any commercial work), it's polyisocyanurate, in various
thicknesses usually between 1" and 3", sometimes tapered for drainage.
It's not entirely unlike styrofoam, actually probably more like the
dried end result of aerosol insulation in a can. It has a fibery paper
face that's less irritating than MDF.

I digress. Besides being thermal insulation, that stuff is used as a
gentle, flat, supportive base on which to glue the EPDM. Additionally,
insulation is easy to glue down using the same contact cement as the
EPDM, or if you screw it down, you can screw through plates that
protect the EPDM from the screw heads. Liquid Nails and Luan is a
decent solution, and may be common in RVs for all I know.

Their duct taped seams and edges on the Luan are not confidence
inspiring for me. Duct tape is pretty thin, and if there's any movement
in the Luan, Bad Thing can happen. If the EPDM is glued well and
there's any movement, the joint will repeatedly stretch the EPDM,
which, after a few years of UV exposure, may not be quite so stretchy.
If it's not glued well, the joint edges will abrade the EPDM and
eventually wear through it.

EPDM does seem very tough as they said, but that's not necessarily so.
Still, the extra 1/2" thickness of MDF might be less than optimum for
an RV, so maybe just some better protection at the joints is called
for. Perhaps the butyl tape, applied to the Luan seams before attaching
the EPDM, would be called for.

Working under cover sound great, too.