On the set of the third season of “The White Lotus,” which shot for seven sticky months in luxury hotels in Bangkok and Koh Samui, Thailand, writer and director Mike White had a repeated note for actor Patrick Schwarzenegger.

“You’re not walking rich enough,” White would yell across the pool deck. “Patrick, be richer.”

Schwarzenegger, 31, recounted this — incorporating an impeccable White impression — on a bright morning at a coffee shop in the Tribeca neighborhood of Manhattan. In person, he was polite, earnest.

“I’m thankful each and every day for the life that I’ve been given,” he said as he spooned up yogurt and berries.

White saw this guilelessness as genuine. “Patrick is just somebody who likes people, and people like him,” White said on a call earlier that week. “He’s a sincere actor, but he’s uncomplicated in his presentation of self.”

Schwarzenegger plays Saxon, the eldest son of a wealthy North Carolina couple (Parker Posey and Jason Isaacs in “White Lotus.” A cocky finance bro, Saxon’s preferred pastimes include smoothies, pornography and observations about his siblings’ sex lives.

“She’s pretty hot,” he tells his brother (Sam Nivola), speaking of his sister (Sarah Catherine Hook). “But I don’t think she’s ever been laid before.”

Schwarzenegger, who glowed beneath the cafe’s lighting, often plays entitled young men. His character in “Gen V,” the college-set Amazon show that is a companion to “The Boys,” is actually named Golden Boy.

That casting isn’t complicated: Schwarzenegger has wide-set eyes, honeyed hair, a high-thread-count affect. That demeanor ensures a resume of jocks and pretty boys. But in his best work — “Gen V,” the HBO limited series “The Staircase” and now “The White Lotus” — he is able to get under the moisturized skin of these young men, showing something darker and more wounded.

“He’s got this Tom Cruise quality where he’s very cheerful, but underneath it there’s this anger,” White said of Schwarzenegger’s onscreen persona.

On the handsome face of it, Schwarzenegger and Saxon are alike. They both joined frats in college. They both studied business. Neither stints on arm day. Even in his downtime, Schwarzenegger does not exactly walk poor.

Schwarzenegger is the eldest son of Arnold Schwarzenegger and Maria Shriver, which means he descends from both Hollywood and political royalty. (Shriver’s parents were Eunice Kennedy and Sargent Shriver.) He doesn’t deny his good fortune or the advantages it brings.

“You would have to be totally out of whack to not understand the privilege,” he said. He’s happy to lend some of that to Saxon but is quick to list the differences between them, which he identified as “his vulnerability levels, his relationships, what he values in life.”

His colleagues noted other differences. Posey, his co-star, said he had a lot of grace about him, a lot of maturity. He was, she said, a very nice young man. White said, “He’s got a gym bro vibe a little, but he’s just a very kind person.”

Like many children with parents in the industry, Schwarzenegger spent much of his childhood on set, doing homework in his father’s trailer, making peanut butter sandwiches at craft services. He wanted to be an actor because that was what his dad did. He had also done school plays and loved the experience, he said, of “totally trying to be someone else.”

While minoring in cinematic arts at the University of Southern California (his father pushed him to major in business) and studying at an outside studio, Schwarzenegger began to get roles — an Ariana Grande video, an episode of “Scream Queens,” a rom-com called “Stuck in Love.” Soon he was hired for “The Staircase.” “Gen V” and “American Sports Story,” in which he played Tim Tebow, followed. Before flying out to Thailand, a 14-hour time difference from his fiancee, Abby Champion, and family, Schwarzenegger worked out an elaborate backstory for Saxon.

The role required nudity and at least one disturbing sex scene. He was OK with it. It was right for the character, he said — so right that he elected to do a scene in the first episode without the boxer briefs that wardrobe had offered. “Whether that’s uncomfortable or weird for Patrick, it doesn’t matter,” Schwarzenegger said.

He’s not sure what his family will think about the more explicit material in coming episodes. But he’s excited for viewers to see that he can really act, that he can make Saxon’s distress real and funny and even sympathetic.