Up at my North end, it's always cooler than any surrounding areas. Down your way, I don't know.
It's always a nice ride (45PH, so it takes awhile), especially you get behind Winnebagos and trailers. I'll pass on a double yellow, if I can see aorund and do it quickly and safely, but you risk a big ticket.
It's always a good idea to call to see if any sections are closed for maintenance (or ask at your entrance gate), or being chip and sealed with nasty hot tar and gravel (which you don't want to ride on.... and scrub your azz off getting off) (I've had to backtrack and go around the long way several times)
https://www.nps.gov/blri/planyourvisit/roadclosures.htmAlso, print this out and carry it with you for gas availability by mile marker. You can rarely actually see gas stations from the Parkway, and sometimes they are right down the hill, and other times they are 20-40 miles from the exit you took. This takes all the guess work and worry out of it. (Though maybe your cell phone and GPS does all that for you).
https://www.blueridgeparkway.org/gas-availability/