The swan is doing one of the following: Resting its leg or trying to keep cool. Swans typically keep one leg up to rest. However, there is research that the swan's leg is placed on its back because the webbing has many blood vessels, specifically capillaries. Dogs pant and yes we have seen swans pant also. But, the major dissipation of heat seems to take place through the blood vessels of the webbing. The Regal Swan