We also use the blood vessels and airways in our heads as heat exchangers. We know that the brain is very sensitive to heat and that excessive heating can cause death. Cats and sheep, for example, have an arterial network on the forehead and as blood passes from the arteries, it cools down to the brain. Humans and primates do not have this network, but the carotid internal arteries that go through the neck to the brain pass near the airways and the cowbell, where apparently the blood cools.
To measure it, microthermometers have been placed on a model and they have seen that the blood of the brain is colder than the rest of the tissues. They say this would be related to the sudden death of children, as falling asleep upside down would not get enough cold air into the neck and airway area, so the brain would die with excessive warming.