Rune, who cruised into the fourth round with a 6-4 6-1 6-3 win over Argentine qualifier Genaro Alberto Olivieri on Saturday, said he needed as much sleep as possible to stay fresh in the tournament.
"I sleep a lot. I love to sleep. I probably sleep around, I don't know, nine to 13 hours if I can," he told a press conference following his third-round victory.
"It's a lot. I really like it. I think it's the best recovery that you can have. You know, I actually think the muscles relax, everything relaxes when you sleep, so I try to, every time I can, to sleep," he said.
One of the fastest-rising stars on the tennis tour, Rune is making his second appearance at the French Open after reaching the quarter-finals in his debut last year.
It is not just sleep that keeps the title-hungry youngster in fighting mode; it is also the dream of a first Grand Slam trophy.
"Sometimes I have very good dreams that, you know, I stand there with the trophy and stuff but then when you wake up, you're like 'Oh, shit', I'm just in bed'," he said.
"I think we all dream, no? But sometimes about tennis, sometimes not. Can be random things.
"But then when you actually make the dreams into a reality it feels even better, because then you remember, oh, that's actually what I dreamt about."
Victory in Paris would make Rune the first men's singles Grand Slam champion from Denmark.
He will next face either American world number eight Taylor Fritz or Argentine Francisco Cerundolo in the round of 16, hoping his dream run in Paris continues.