First, a bit of background.
Vaporizing a liquid requires the input of energy in the form of heat. Just think what happens when you spill a bit of nail polish remover or alcohol on your hand. It doesn’t take long to feel a cooling effect! That is because the heat required to convert the liquid to a gas comes from body heat. This is also the principle behind a refrigerator. A gas, Freon, is compressed by a motor into a liquid, and when the pressure is released, the liquid evaporates, drawing the required heat from inside the fridge. The gas is then compressed again and the cycle continues.
Perspiration is the body’s cooling mechanism. The evaporation of water and sweat is mostly water, requires heat and that heat is taken from the body. But the extent to which water can evaporate from the surface of the skin depends on several factors. The greater the extent to which the skin is covered with clothing, the greater the impediment to evaporation. If there is not much clothing, then some of the sweat may drop to the ground before it can evaporate and then, of course, it cannot take part in the cooling process. But probably the most important factor is the amount of water vapour the surrounding air already contains, which is the “relative humidity,” defined as the percentage of water vapour present in the air relative to the amount needed for saturation at the same temperature.
Drinking a hot beverage stimulates nerves in the mouth and on the tongue to send a signal to the brain that there is incoming heat and that a cooling mechanism should kick in to maintain body temperature. The response is perspiration. However, if the surrounding air is already saturated with water vapour (100% relative humidity), then the secreted moisture will not evaporate readily and there will be no cooling. If, on the other hand, the air is dry, then drinking a hot beverage can indeed have a cooling effect. Incidentally, spicy foods can also stimulate sweating, possibly explaining why such foods are often popular in hot countries. In any case, when it comes to perspiring a lot, the most important thing is to replace moisture loss to prevent dehydration. As far as deciding whether that should be with a hot or cold beverage, well, don’t sweat it.
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