Rock drummer Carmine Appice co-wrote a No. 1 song, Rod Stewart’s “Da Ya Think I’m Sexy”; played with everyone from Jeff Beck to Pink Floyd; befriended his idol, drum legend Buddy Rich; and even, to hear him tell it, influenced Led Zeppelin.

But one of his stranger achievements was patrolling hotel hallways as a member of the “Sex Police.”

The group — consisting of Appice, Stewart and other members of Stewart’s band — would listen for anyone in the band or crew having sex and sabotage these encounters any way they could.

After donning their official “Sex Police” T-shirts, the crew would “charge down the hotel corridor to that person’s room, singing our theme song as we went: ‘Sex Police, we’re the Sex Police. Sex Police, we’re the Sex Police!’ ”

Once inside the room, they would “cause as much mayhem as possible. Beds might get tipped over; sometimes the chick would be thrown out of the room without her clothes. Normally she would take one look at the 10 to 15 guys bursting into the room and leave of her own accord anyway.”

Having begun his career in the ’60s, Appice relished rock ’n’ roll at its most depraved. In “Stick It!,” his new memoir, he makes it clear that having sex with groupies and trashing hotel rooms, often alongside the biggest rock stars of the era, was his raison d’être.

Jimi and Zep

Born in Borough Park, Brooklyn, in 1946, Appice fell in love with the drums in his teens.

One early band, Thursday’s Children, played dingy, junkie-filled blues clubs on the then-decrepit Upper West Side, where they were once booked to play with an outrageous young guitarist named Jimmy James. Appice and James — who would soon become Jimi Hendrix — became friends while getting high in a local prostitute’s apartment.

It was with this band, in 1964, that Appice learned the sexual power of rock. When he began playing with them at 17, he was a virgin. By the following year, he claims to have slept with over 300 women.

“As the band started playing shows most evenings, it became normal to have a different chick every night of the week. Sometimes it was two,” he writes. “I would have a quickie with a girl in the back of the club, then another one would take me back to her apartment after the show.”

Three years later, his band the Pigeons was signed to Atlantic Records, changed their name to Vanilla Fudge, and had a big hit with a cover of the Supremes’ “You Keep Me Hangin’ On.”

‘The four of us [Vanilla Fudge] disappeared into the bathroom, emerging butt-naked. A couple of girls freaked and ran out of the room. We had an amazing sex party with the girls who were left.’

Appice was now a star, and a West Coast tour indoctrinated him into the era’s groupie scene.

In LA to play the Whisky a Go Go, he casually announced during a radio interview that the band was having “the mother of all parties” at their hotel. When they returned, they found 500 fans waiting for them in the lobby.

“We had a rather more exclusive party in mind,” he writes, “so we picked out a bunch of the hottest chicks and took them up to our floor.”

They brought all the girls to their hotel room and decided on a “short cut” to see who was “up for sex.”

“The four of us disappeared into the bathroom, emerging a minute later . . . butt-naked. A couple of girls freaked out and ran out of the room. Most of them didn’t. Let’s just call it a process of natural selection. We had an amazing sex party with the girls who were left.”

One of Vanilla Fudge’s early opening acts was a new band called Led Zeppelin. While guitarist Jimmy Page and bassist/keyboardist John Paul Jones, both studio pros, were somewhat well known, singer Robert Plant and drummer John Bonham were barely off their local pub circuit.

During their first show together, in December 1968 in Denver, Zeppelin was booed. Plant “stood stock-still onstage; I even took it upon myself,” Appice writes, “to tell him he should move around a bit more.”

Appice formed a long friendship with Bonham, whom he describes as “the nicest, kindest, gentlest guy to be around — until the demon drink passed his lips.”

Bonham died in 1980 after consuming 40 shots of vodka in a 12-hour span. According to Appice, rumors flew that he might replace Bonham in Zeppelin, and these rumors played a part in the souring of his relationship with Stewart, who felt they were distracting from his music.

The mud shark

Led Zeppelin and Vanilla Fudge toured again a year later, this time as co-headliners, as Zeppelin’s star was rising fast. It was on this 1969 tour, in Seattle, that both would participate in the most notorious groupie incident in rock history.

Appice was there, and gives a full, disgustingly detailed recounting of the infamous Led Zeppelin mud-shark story, wherein Bonham and/or Zeppelin’s tour manager, Richard Cole, have long been said to have pleasured a young woman with a shark.

According to Appice, the story is not only true but much more intricate and debauched than most tellings have conveyed.

The Edgewater Inn, where both bands stayed, was right on Elliott Bay in Puget Sound, which allowed guests to fish from their hotel-room windows.

As Appice tells it, he, his bandmate Tim Bogert, Jones and Plant and Plant’s wife, Maureen, were getting high in Jones’ room when there was a knock at the door. After they scrambled to hide their pot, it turned out to be a girl Appice had had a series of dalliances with (described in minute detail in the book) the day before. Very high, the girl — in response to something Appice had told her about his bandmate having a movie camera — kept telling the others she wanted to make a movie with them.

Appice, too high to deal with her, went next door to hang out with Bonham and his wife, Pat, Cole and two crew members, several of whom were fishing out the window. They had a “mini-aquarium” in their bathtub, including “a 2-foot-long, dead-eyed, ferocious-looking mud shark.” He told them about the girl, then later returned to the first room.

There, Appice and the rest listened to music as again there was a knock at the door. When they opened it, Bonham, Cole and the two crew members barged in, with Appice’s bandmate, Mark Stein, “holding a Super 8 camera and lights.”

From here, Appice writes, it was “carnage,” as Bonham and Cole went right for the girl.

“ ‘You want to make a f—ing movie?’ they asked her. ‘OK, let’s do it right now! Take your clothes off!’ ”

As she did so, Appice realized that the camera and lights weren’t all they had brought from the other room. They also brought the shark.

Cole and a crew member threw the girl on the bed. With Stein filming, Cole took the shark by the tail and “started whipping the girl with it, beating her again and again as she writhed around the bed.”

Each time the shark made contact, Appice writes, “its teeth ripped her skin and left tiny blood-red scars all over her back.” Throughout, the girl “was bucking and screaming with pleasure,” as if in the throes of orgasm.

Soon, the shark was being used differently, having sex with the girl as Appice and the others “were all doubled up with laughter as the insane scene developed.” The noise attracted the attention of the hotel manager, who burst into the room, saw the scene, and demanded they stop at once. Appice, Jones and Bogert scrambled out of the room in hysterics.

This is only the first part of the story. The second part is even more depraved, as roadies used the woman and humiliated her further. Jones finally had to leave in disgust, and Appice says he would have done the same had it not been taking place in his room.

The next day, Appice saw Frank Zappa at the airport and shared the story with him, and Zappa immortalized the tale in a 1971 song called “The Mud Shark.” (In the song, Zappa says the story was told to his keyboardist, Don Preston.)

The film, most likely to everyone’s relief, never surfaced. Years later, the girl called in while Appice was doing a radio interview, telling him she had raised a family and was living in Alaska.

4,500 to 1

Appice played in many bands over the years, both his own and others. Vanilla Fudge was followed by Cactus, then by Beck, Bogert and Appice, in which he had many disputes with guitar icon and on-again/off-again friend Jeff Beck.

He played in Ozzy Osbourne’s band for a time but writes that Osbourne’s manager/wife Sharon was angered by increasing attention he received for his special-effects-laden drum solo on the tour.

Previously told he could sell his own merchandise, he arrived for one show to find that his face had been cut out of his T-shirts — on every single shirt. When he asked who had done this, the answer was, “Ask Sharon.” That night, as he began his solo, his special effects failed. He was fired from the tour soon after.

Sometime later, he was using a friend’s spare room in LA as a work base/crash pad. The friend worked for his management company, and a young artist new to their roster moved into the room full-time.

“His name was Prince,” Appice writes. “And he was funky.”

Prince was “very quiet and introverted,” and when he spoke, it sounded like “somebody was whispering at me from two miles away.”

Their relationship was cordial. Once, Appice walked into the room and found Prince in bed with Vanity. He quickly apologized and backed out.

Appice was also there when Prince opened for the Rolling Stones at LA’s Memorial Coliseum.

‘Leslie asked me a loaded question, and the shocking answer was 4,500. But number 4,501 was the one that mattered most.’

“The Stones fans didn’t know what to make of his stage gear of stilettos, stockings and panties and booed him off,” Appice writes. “I was at the house when Prince came home freaked out and in tears.”

Appice and their mutual friend tried to calm him down, and Appice shared some of his own disastrous experiences, as when Vanilla Fudge bombed when opening for the Mamas and the Papas. It was to no avail. “Prince was inconsolable.”

Appice was married, to five different women, for much of his groupie-loving existence. Now, he has been in a relationship with a New York rock DJ named Leslie Gold — known as Radiochick — for 13 years. At one point, Gold insisted on figuring out how many people he’d had sex with in his life. Her own total was seven.

Using a spreadsheet, she arrived at the number 4,500.

Now 69, Appice insists he has always been faithful to Gold.

“Leslie asked me a loaded question, and the shocking answer was 4,500,” he writes. “But number 4,501 was the one that mattered most.”