Grape - covered or not?

If you are doing a long run and it is hot out (>85F) and very little shade, is it better to wear a full white cap or just wear a visor? In the military, you were always taught to cover your grape when in the sun. But a cover would obviously hold in some heat. What is the current thinking on this issue?

Disclosure - I still wear a light weight runner’s hat with the mesh on the sides, under the conditions described above. Similar to this one http://www.roadrunnersports.com/rrs/products/ASW1133/

Take a look at this thread.

The short answer is it depends on the ambient sunshine, humidity, and wind; as well as the color, air permeability, and dampness of your cap.