The answer is they don't sweat to cool down. They have lots of blood vessels in their big ears and by exposing the blood to the cooler air, cool their bodies. For all those who are suffering in the current heat wave here is a solution that uses the animal's wisdom.
For some time I have made cool scarves for service persons in Afghanistann and Iraq. The crystals absorb a great deal of water and hold it for a long time. As a result the scarves when draped around the neck, through evaporation, cool the blood running through the veins and arteries in the neck. If you cannot cool the the environment, cool the body. The scarves are easy to make, requiring just a narrow fabric tube and a bit of the crystals. In addition extra scarves can be kept in the refrigerator for an instant whoosh of coolness. The scarves can be refreshed as they lose moisture by a simple soaking in water. This might be a good project to make for seniors who do not have air conditioning and cannot go out to a cooling center. It might also be good for those who have jobs requiring them to work outdoors. At a minimum it can make a hot day more bearable at a lot less cost than the energy to run an air conditioner on high cool. Here is a link for instructions and sources of supply.
http://soldiersangels.org/uploads/CoolScarvesHowTo.pdf