specular and diffuse reflection

I'm not exactly sure what the described situation is (is there a coating on the surface of the foam board being increased/decreased, or maybe the idea is that different foam materials can be used which would change the reflectance of it?).

I think specular reflection is just for smooth surfaces like mirrors so that you can see some sort of reflected image. A typical foam board would cause only diffuse reflection. I guess the rougher the board's surface is, the greater the chance that some radiation could reflect but still move into the board giving it another shot at contact. If it's a probability that the foam material will absorb the EMR with each contact, then a rougher surface would cause a (probably small) increase in the number of contacts, which could result in a little more absorption.

Any electromagnetic radiation (EMR) that enters the board would 1) reflect, 2) be absorbed, or 3) pass through. (It might penetrate some before being reflected or absorbed.) I don't know how foam interacts with the various frequencies of EMR.

In the cases of 1) reflection and 3) pass-through, the EMR remains as light energy, whereas with 2) absorption it gets absorbed by the electrons (I think) of the foam and is converted to thermal energy, which would then be conducted through the board towards lower temperatures.

You might feel heat from either the cases of 2) absorption, because the board gets hotter, or 3) pass-through if the electromagnetic radiation (EMR) warms your hand directly. You can eliminate the possibility of the EMR warming your hand directly by removing the board from the EMR and testing its temperature in the shade.

If you were to increase the reflectance by using a more reflective foam (like white instead of gray), somehow polish the board, or spray on some reflective material, then it would reflect more of the EMR so the foam board wouldn't gain as much heat. And it all depends on what the materials are, what frequencies of EMR there are, and how they both interact.