I'm betting your drummer doesn't put his kit on the asphalt either.  Probably has a rug, at minimum, if not a riser.

This is why I put my MFX boards in pedal flight cases.  They never touch the ground, stage, etc, directly.

The styrofoam sounds like a pretty smart option to me.  Preferrably white  

I think Karl has given excellent advice by choosing to use a pedalboard style carry case that fits all of your floor gear not just the POD HD.

If this is not an option, then thick cardboard and heavy duty styrofoam will be a decent solution. 

For the truly budget conscious - you can make an insulated "floor" for your floor based gear.  try to get as much of this for free as it is not valuable and many food stores, hardware stores have scrap pieces they just throw out you could ask for.

Here's what you do...

1. Get the heaviest duty cardboard box or pieces you can put underneath your entire gear area.
2. Get heavy duty (1" thick) solid styrofoam in 12" x 12" squares or one large piece you can cut down with a box knife or something equally as sharp for smooth edges.
3. Buy THIS specific fan for about $40.00 made by LASKO which will also serve as a two outlet extension cord for your POD HD unit and whatever else.  It can handle up to 10 amps.  Way more power than ever needed by any FX pedalboard.  The 3-speed fan is very quiet yet very high power and insulted from sending stray power distortions into your gear.  The fan itself can be spun all the way around so it is blowing air along the floor.
4. Once you've gotten the cardboard and the styrofoam laid out so that you have a footprint of your pedalboard floor coverage area.  Cut out or put pieces together whichever you need to do to built a bit of a vented deck for your gear to sit on.
5. Like a sandwich.  Cardboard bottom, 1" thick styrofoam, then cardboard top.  Spray paint if you feel the urge to be colorful.
6. Then at your gig, put the cardboard "stage" for your pedalboard gear, place this LASKO fan next to it with the vent pointed at your gear and if possible keep all power supplies on the cardboard - at least the first layer so that the fan will cool your power supplies as well as your processing gear.
7. If you feel like you'll pass out from the heat, please make certain you have plenty of water or Gatorade.  Try not to drink too much alcohol based fluids as this will make your insides heat up faster than you will notice.  Juice, soda are all not the best for very hot weather situations.  I perform outdoors in Florida all year, take my word on this for those who might not have ever passed out from the heat.  It's way more serious than you might think. 
8. Keep an eye on those in your band, especially your drummer.  make certain all all keeping hydrated properly and are sweating.  If you see anyone or yourself stop sweating during your performance, take a break ASAP and get yourself some ice and put it on your head until you start sweating again.  No sweat, & clammy cool hands, in high temps are the first signs of heatstroke.



I hope this helps answer the question about heat and your gear, plus some caution for those not used to playing in very hot weather.

Take care,

Neal

PS - For the guy with the gig inside a Volcano - I think anything less than fireproof gear to wear and fireproof cases are your only option.