This speculation is borne out by a more recent study performed by researchers at the University of Cape Town South Africa. In this study, cyclists performed a time trial in a hot environment on several occasions, consuming fluid at a different rate in each. The authors of the study found that the rate of fluid intake had no effect on core body temperature, but it did affect performance. The cyclists performed best when they drank at an "ad libitum" (freely chosen rate).
So while drinking while running in the heat will not cool you down, it will speed you up. Specifically, drinking during hot-weather runs will keep your blood volume at close to normal levels, which in turn keeps your sweat rate high. And since oxygen is delivered to the muscles through the blood, maintaining your blood volume through drinking also enables your heart to deliver more oxygen per contraction, so you perform better than you can if you allow your body to become too dehydrated.
More: How to Create a Race-Day Hydration Plan
How much should you drink? Studies such as the one above suggest that you should simply drink according to your thirst. Drinking more will neither keep you cooler nor improve your performance; but it will increase your chances of suffering from GI distress.
By far the most effective way to prevent your body from overheating while running in the heat is not to drink a ton of fluid but simply to slow down. But your brain, through its anticipatory regulation mechanism, will strongly encourage you to do this anyway, at first by making you feel uncomfortable at your normal pace and then, if necessary, by simply refusing to allow your muscles to work as hard as you want them to.
More: Hydration Basics
This mechanism is no failsafe, however. During exercise in the heat, it is possible for the brain itself to overheat, causing this protective mechanism to fail and opening the door to heat illness. So, to avoid this dangerous situation, take all the usual precautions such as avoiding exercise during the hottest part of the day, wearing appropriate technical apparel, and heeding warning signs such as dizziness, light-headedness and cessation of sweating.
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