Welcome to the forums Peter
Is your forum name your real name as well? If so I would have thought you might be able to tell us more about it than we could
I once read a paper outlining a number of theories related to the evolutionary fitness benefits of the zebra coat. The main theories they looked at were that the coat evolved as a predatory avoidance characteristic, making the zebra look bigger, as well as the fact that stripes blend in with tall grass. Also along these lines is the idea that moving stripes dazzle predators and that the colouration and patterning are hard to pick out in poor light.
The next major theory looks at the evolutionary development along the lines of social benefits ITO group bonding as a marker for grooming and as a means to identify individuals in the group.
Then comes the idea of thermoregulation which is basically the question you have asked, but unfortunately from an evolutionary and therefore fitness point of view this theory has little merit.
Then lastly my favourite theory. Where the coat is believed to have evolved as a means of protection from tsetse flies.This theory has quite allot of evidence to support it and makes logical sense too. Waage showed experimentally that there was a significant difference between the number of flies attracted to solid colours and those attracted to striped patterns (horizontal).
All of the theories have their advantages and disadvantages and I would list them but a bit short of time. If anyone has access to the paper the reference is:
Ruxton, G.D. 2002. The possible fitness benefits of striped coat colouration for zebra.
Mammal Review 32(4):237-244.