Not quite. This idea comes from a 1973 study that appears to have had some experimental issues with the setup. I am trying to find a more reliable source to debunk this than Io9, but everything so far is coming up behind paywalls or dead links. Until such time, here are the main critiques of that study and some thoughts on it from a more modern look (emphasis mine):

They concluded that the animals stored metabolic heat — that is, they didn't lose any heat from evaporative cooling and instead kept it all in. When cheetahs run, the heat quickly builds up, capping off at 40.5 degrees C (104.9 degrees F), at which point further exercise becomes thermally impossible.

"As soon as their temperature got to 40.5 degrees, they refused to run," Hetem tells io9. "The authors tell stories of how the poor cheetahs would fall off the treadmill."

In the experiments, the cheetahs ran for about 2 km (1.2 miles) at an average speed of 10 km/h (6.2 mph), with maximum speeds of up to 30 km/h (18.6 mph). Based on their measurements and calculations, the authors surmised that a cheetah would stop sprinting after about 30 seconds during a real hunt — 30 seconds is the actual average hunting time of a cheetah in the wild. The conclusions made sense at first glance, but there's a major issue with the study: It didn't simulate real hunts, where cheetahs will sprint at up to 100 km/h (62 mph) for only several hundreds of meters.

So instead of running on treadmills, they decided to put temperature sensors on cheetahs in the wild and observe them hunting, and they found some interesting results:

During the study, the cheetahs' had an average daily body temperature of 38.3 degrees C (100.9 degrees F), with minimum and maximum temperatures of 37.3 degrees C (99.1 degrees F) and 39.5 degrees C (103.1 degrees F), respectively. Cheetahs, like most other mammals, humans included, have a 24-hour rhythm of body temperature, Hetem says. The researchers found that the cheetahs' temperatures didn't rise significantly during the hunts. Indeed, their temperatures at the end of both successful and unsuccessful hunts averaged 38.4 degrees C. The main factor affecting terminal temperatures was the time of day when the hunt occurred — that is, their body temperature when the hunt started, as regulated by the 24-hour temperature rhythm.

The surprising part was that the temperature rises didn't occur until after the hunt, and were more marked after a successful hunt:

The team did find, however, that the cheetahs' body temperatures began to slowly rise after the hunt was over. Surprisingly, successful hunts resulted in a body temperature increase of 1.3 degrees C (2.3 degrees F), while unsuccessful hunts only increased body temperatures by 0.5 degrees C (0.9 degrees F). Furthermore, the cheetahs' temperatures seldom hit the supposed thermal exercise limit of 40.5 degrees C.

They do note this may be to other factors:

The more likely cause of the increase is "stress hyperthermia." Among big predators, cheetahs are near the bottom of the hierarchy. After hunting, they are very tired and most vulnerable to other big cats, such as leopards and lions. And if they have a kill, it becomes even more likely that another predator will come along, so the animals must remain vigilant after hunting — cheetahs have previously been described as being nervous after a kill and alert while feeding. They experience hyperthermia because they are stressed out, similar to the way that antelope experience fear-induced hyperthermia.

Additional problems with the 1973 study are listed as:

On the other hand, the temperature increases seen in the 1973 study may have been due to the setup of the experiment. The animals were running for long periods of time — on a treadmill — and were getting their temperatures taken rectally, an experience which may well have been stressful.

So, why do they give up after about 30 seconds? No one is sure as of yet (according to my single source still):

There are several theories floating around, Hetem explains. The most probable explanation: They ran out of energy. Sprinting is a very anaerobic exercise that results in the build-up of lactate acid, which disrupts the breakdown of glucose to energy. "The cheetahs may just run out of energy after 30 seconds of sprinting," Hetem says.

So, I wanted to put this out there, but will come back to this in a bit if I find more sources to corroborate.

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