An unseasonal stretch of extreme heat and humidity has left the stadiums at the Open sweltering. But a few players, Coco Gauff among them, say the hotter, the better.In most years, there is a very specific climate pattern at the U.S. Open.The tournament starts at the end of the dog days of August, in the lingering heat and humidity of a New York summer. By the final matches, at the end of the first full week of September, it’s a good idea to bring a light sweater or a windbreaker to the Billie Jean King National Tennis Center.Not this year. Not even close.A first week filled with cool, breezy afternoons and crisp nights has given way to some of the hottest days — and nights — of the summer, with conditions that have brought some of the fittest athletes in the world nearly to their knees, even when they are playing in twilight and after sunset. It is heat and humidity so oppressive that it parks itself in the brain, sparks fear and makes it difficult to focus on anything else, especially returning serves of 130 miles per hour and chasing forehands and backhands around the court for as many as five hours.Earl Wilson/The New York TimesIt is the first thing that Daniil Medvedev has been thinking of when taking the court for his warm-ups this week, sessions that take place hours before his matches.“I was like, ‘Oh, my God,’” Medvedev said the other day as he prepared to play Alex de Minaur of Australia. Medvedev is from Russia and, like many of the Eastern European players, can become awfully cranky in extreme heat.In a quarterfinal match on Wednesday, he struggled to see the ball and relied on instinct to survive a grinding battle with his countryman and close friend, Andrey Rublev. For the second consecutive day, organizers used a new measure to bring relief — partially closing the roof of Arthur Ashe Stadium to shade the court.“One player gonna die, and they gonna see,” Medvedev muttered in the middle of the match.Even still, after Medvedev prevailed in straight sets in two hours, 47 minutes, he slumped on his chair, draping a towel packed with ice around his neck, his head between his knees, begging for water. Had the match stretched to a fourth set, Medvedev said he would have used the 10-minute break to take a cold shower, even though he knew it might make his body stiff as a board.Gabriela Bhaskar for The New York Times“I didn’t care, I was going for the shower,” said Medvedev, the skin on his face raw hours later from rubbing it with a towel too much.“Brutal,” is how Cliff Drysdale, the longtime tennis commentator for ESPN, described the afternoon.As the planet warms, officials in every warm-weather sport are searching for a balance between safety and maintaining the belief that elite sports demand elite fitness and the ability to win in challenging conditions. International soccer has incorporated water breaks in extreme heat. Track and field has started scheduling marathons at dawn or at night.Tennis, which has become more physical and taxing during the last 20 years thanks to improving racket and string technology and court conditions, is navigating the issue as well.“It’s part of the sport,” Stacey Allaster, the tournament director for the U.S. Open, said of the heat.Frances TiafoeAmir Hamja/The New York TimesBen SheltonAmir Hamja/The New York TimesHiroko Masuike/The New York TimesTennis players are not strangers to extreme temperatures. Their seasons begin in the Australian summer in January, where hot winds from the arid plains can send temperatures into the triple digits and make the tournament feel as though it’s taking place inside an oven. At the Australian Open in Melbourne, shifting winds and temperature swings of 20 to 30 degrees within a few hours are not uncommon.After Australia — though there are a handful of indoor tournaments — the sport essentially spends the next 10 months chasing the sun. There are steamy stops, such as Doha, Dubai, Florida, and Mexico; and even August events in Atlanta, Washington, D.C., and outside Cincinnati ahead of the U.S. Open in New York’s “big heat,” as Novak Djokovic refers to it.This week, that heat has been very big indeed, requiring Allaster; Jake Garner, the tournament referee; and their team of advisers to keep a close eye on the WetBulb Globe Temperature, a measure of the heat stress in direct sunlight, which also takes into account temperature, humidity, wind speed, sun angle and cloud cover.When it rises above 86 degrees, mitigation measures kick in, including the 10-minute break between the second and third sets of the women’s matches and the third and fourth sets of men’s matches.Garner said in an interview on Wednesday that officials this summer decided that when the index hit 90 degrees, he and his team would meet to consider whether to partially close the roofs at its two main stadiums, Louis Armstrong and Arthur Ashe.It crossed that threshold on Tuesday, nearing 92 degrees on the court during Coco Gauff’s quarterfinal win over Jelena Ostapenko. Had that match gone to a third set, the roof would have been partially closed, but Gauff won in straight sets. So officials shaded the court for the next match, Novak Djokovic’s straight sets win over Taylor Fritz.“We both struggled,” Djokovic said. “A lot.”On Tuesday, temperatures became so hot that officials shaded Arthur Ashe Stadium during Novak Djokovic’s quarterfinal victory over Taylor Fritz.Amir Hamja/The New York TimesLater in the afternoon, on one of the field courts, Stephane Houdet, who is participating in the wheelchair tournament, stashed a water bottle in the box near the baseline where players keep their towels, sipping from it between points.“A great idea,” said Brian Hainline, the chairman of the United States Tennis Association, who is a physician and the chief medical officer for the N.C.A.A. The problem for the U.S.T.A. — and, ultimately, the players — is that even with the roofs closed, both stadiums are designed as open-air venues that cannot be sealed. They have air circulation systems that prevent moisture from settling on the court when the roof is closed, rather than fully operational air conditioning systems. On the bright side, the complex is just a stone’s throw from Flushing Bay, and when there is wind coming off the water, it can be cooler there than in many spots in New York City. Unfortunately, the wind has been lifeless in recent days.As players booked their spots in the semifinals set for Thursday and Friday, there seemed to be a clear pattern emerging — Florida. Two of the three women who had made the final four by late Wednesday afternoon, Gauff and Aryna Sabalenka, make their homes there. A third, Madison Keys, who lives in Orlando, claimed the final spot on Wednesday night with a 6-1, 6-4 win over Marketa Vondrousova of the Czech Republic. Ben Shelton, the 20-year-old with the cannon serve who will play Djokovic in the semifinals on Friday, lives in Gainesville, Fla.Aryna Sabalenka said training in Florida helped her cope with the heat during her quarterfinal win over Zheng Qinwen.Earl Wilson/The New York TimesSabalenka, who grew up in Belarus, hardly a tropical locale, credited her summer training near her home in Miami as she managed to resist wilting in Wednesday’s heat during her win over Zheng Qinwen of China.“What can be worse than Florida?” Sabalenka said.For Gauff, the 19-year-old from Delray Beach, Fla., who has become the darling of the tournament, the heat represents an opportunity to thrive rather than something to merely survive.“The hotter the better,” Gauff, who will face Karolina Muchova, of the rarely hot Czech Republic, on Thursday, has said on more than one occasion.That may be especially true against Muchova. She struggled against Gauff in the Ohio heat last month during the final of the Western & Southern Open. She walked onto the court for the warm-up that day, and said, “Oh, Jesus.”“Ouch,” she said when it was over.On Wednesday, one of Muchova’s coaches, Jaroslav Blazek, said he would have her focus on trying to keep her body cool. Many players have been sticking black hoses that spray cold air under their shirts during the changeovers. But he anticipated the challenge would be as much a mental battle as a physical one.“You should be ready that it’s going to be like in hell,” he said. More