The thermometers read 30C in Rome on Sunday, but it felt a hell of a lot hotter out on the parched grass of the Marco Simone Country Club. The sky was pitilessly clear, there wasn’t enough breeze to ruffle the leaves, and bare hills gave precious little shade, so the spectators were all pressed together in the lee of the pine trees.

Beyond the shadows it was weather for mad dogs and Englishmen out there, and even they were wearing protection: baseball caps, baggy caps, bucket hats, berets, bandanas, panamas, stetsons, trilbys, even the odd gladiator’s helmet.

There must have been 6,000 people around the 16th green; almost everyone of them had something on top. Inside the ropes Justin Rose had a cap, so did the two caddies, Mark Fulcher and Joe LaCava, the cameramen, the commentators and journalists, the wives, siblings, children, friends and agents, the coaches, the officials, even the kids carrying the scoreboards. And then, out there on his own, alone, was Patrick Cantlay, standing, squinting, over a 50ft putt while the sweat dripped down his forehead. No hat Pat, the last bareheaded man this side of the Tiber.

They will look back on this week and wonder when, why and how the biggest story of the tournament ended up being the curious case of Cantlay’s missing hat. It all started on Saturday afternoon, when Sky’s golf correspondent put out a tweet saying that his sources had told him Cantlay was angry that the team weren’t being paid to play here by the PGA this week, and that he was “demonstrating his frustration” by “refusing to wear a hat”. And it is true, apparently, that Cantlay is angry that someone else is profiting off his golf this week.

Cantlay denied the point about the hat, though. He insisted that the reason he didn’t have one on was because “it didn’t fit”. To his credit, he managed to keep a straight face while he said it. If it was a protest, it’s not exactly going down alongside Tommie Smith’s glove or Henry Olonga’s armband. They’ll remember the crowd’s reaction to it, though, and all the carry-on from Saturday afternoon when they were heckling him so badly that LaCava snapped after Cantlay holed a birdie putt to put the US 1 up in the fourballs match against Rory McIlroy and Matt Fitzpatrick.

LaCava started striding around the green brandishing his cap like an old man shouting off a group of teenagers who were crossing his lawn. McIlroy told him to clear off, since he had a putt of his own to get on with. LaCava came back at him, then Shane Lowry weighed in, while in the background all the US team were waving their own caps around in celebration. And then there were the afters, when McIlroy and Jim “Bones” Mackay ended up shouting about it in the car park that evening, carrying on like it was turning out time down the Dog & Duck.

You couldn’t miss Cantlay on Sunday. The crowd wouldn’t let you. He was serenaded everywhere he went, “Where’s your hat at?” to the tune of Basement Jaxx’s Where’s Your Head At on the 2nd; “Hats off for your bank account!” to the tune of the Village People’s Go West on the 3rd; “Where’s your hat? Where’s your hat? Cantlay! Cantlay!” to the Cranberries’ Zombie on the 5th; “It’s all about the money, money, money!” to Jessie J’s Price Tag on the 7th. In between all that every joker in the gallery was calling out at him or waving cash in his face.

“That’s the money shot, Patrick!” and “Cantlay you should be paid by the hour!” and, from one pithy Scotsman, “Nae wonder you’re no’ getting paid!” when he dumped his tee shot in a bunker at 7. And everywhere he went he was greeted like George VI walking on to the pitch at a cup final, everyone madly doffing their caps, or waving them in the air.

Cantlay didn’t give a damn about any of it. More often than not, he met it all with a smile and a nod, then set himself for his shot. He was happy to be the villain, is used to it even, given all the stick he gets for his slow play.

And he played brilliantly. Too well for Rose. Cantlay went 1 up with a birdie at the 2nd and never lost his lead; it became 2 up when Rose made a bogey at 7, and 3 up with another Cantlay birdie at 11. He was strong enough to hold off Rose when he came back at him with back-to-back birdies on the 12th and 13th.

LaCava, on the other hand, cannot have enjoyed the round. “You having a putt as well, Joe?” the crowd cried out and “When’s it your turn?” and “Watch out for those sight lines, bagman!” But given the way he had carried on the previous evening, he had it coming. Fulcher gave him a lesson in how to do it when he strode out to hush the crowd down before Cantlay’s putt to halve 16.

Cantlay may not love the Ryder Cup the way some of the other players do, he may think that the PGA ought to offer the players more than a $200,000 donation to their favourite charity to play in it, but the sport is big enough for all sorts, and his grit and good humour helped make this tournament. By the end of the round you felt he had even begun to win the crowd around. And the image of him, leaning on his club by the side of the fairway, screwing up his eyes so he could stare down the sun while he plotted his next shot, will be one of the few the US team want to remember from this week.