The Lollapalooza festival at Hoppegarten racecourse outside Berlin: in preparation for their headline slot, Foo Fighters have set up camp in the backstage area. "Help yourself to beer, food, whatever you need," says Gus Brandt, the Californian band's tour manager since 1996, as roadies shift flight cases, statuesque blonde women loll about on sofas and the six musicians run through a handful of songs inside a windowless portable building. All is calm and functional - civilised, even - which, given that one of the biggest rock bands will soon be delivering a thunderous set in front of 100,000 excited Germans, is not what you might expect.

"The fish starts stinking from the head," says Dave Grohl, Foo Fighters' singer, guitarist and undisputed leader, on how to maintain good relations within your extended employee base. On three hours' sleep, Grohl and the drummer Taylor Hawkins are talking to me the morning after the festival in their suite at the Waldorf Astoria in Berlin. While Hawkins comes across as an amiable surfer dude, Grohl, in glasses and black jumper, looks and sounds almost professorial.

"Most of these people have been with us for 20, 25 years," Grohl says. "From the outside it's just a platoon of people in black sweatpants pushing cases around, but the relationships within are like a family and just as complex."

"I can't understand those bands who clean house and sack everyone," says Hawkins, who joined Foo Fighters in 1997 and, like his bandmate, is a picture of American health: lustrous hair, muscular drummer's arms, impressive dentistry. "As much as I'd like to punch Gus in the face every now and then, the thought of doing this without him, or any of the crew, is horrifying."

All of this fits into Foo Fighters' image as the nice guys of rock, a safe pair of hands with whom to enjoy big, festival-friendly songs that will make you feel good without frightening the horses. As Grohl explains, though, this is a band who learnt the hard way. Grohl was 20 when in 1990 he joined Nirvana, who at the time were an underground punk trio from Seattle. A year later Nevermind became the album for disenfranchised youth the world over, going on to sell 30 million copies. Three years after that their lead singer and guitarist, Kurt Cobain, shot himself, leaving Grohl, at the age of 24, with no choice but to pick himself up and start all over again.

"We get a bad rap for being a safe dad-rock band, but don't forget where I came from," says Grohl, who is more softly-spoken than his onstage roar suggests. "I was in the band where heroin addiction was a problem, where one of the members committed suicide, and that is the last place I want to be. I'm a musician. I love playing music for people, and what happened in Nirvana was horrible. And it all happened so quickly, nobody had a chance to take a deep breath and think about what was going on.

"This time we're a little more concerned with the health and safety of each other, and about making business and relationships work, because I can't do anything else and I don't want to go back to working in a pizza restaurant."

"I was that guy for a while: the world's clown," says Hawkins, who overdosed on heroin in 2001, ending up in a two-week coma. "The thing about living fast and dying young is that the person doing it doesn't get to enjoy it. It's everyone else who enjoys seeing what a fuck-up they are."

Recent years in Foo Fighters have not been plain sailing. In 2015 Grohl fell off a stage in Gothenberg and broke his leg, forcing the band to cancel a headline spot at Glastonbury (they returned to do the festival this year) and for Grohl to play out the rest of the tour from a throne, his leg in plaster. After 60 shows he and the rest of the band were, to use Grohl's apt term, wrecked.

"All I wanted to do at the end of that tour was go away," Grohl says, laughing. "Emotionally, physically and mentally we were exhausted. We had to stop this band before we killed it, so we took a big, long break, which we had never done before. I went into my Brian Wilson period, not leaving the house for weeks on end, growing a big beard and refusing to change out of my pyjamas. I was lacking inspiration. Eventually, knowing we needed an album that gave us a reason to remain a band, I went into my home studio and started fooling around. And I hit a vein."

The result is Concrete and Gold, the best thing Foo Fighters have done in years. It combines the kind of rhythmic, riff-based rock for which the Foos are known with sweet Beatles-like melodies and harmonies, which makes sense when you discover that one Paul McCartney - who helped Grohl to recover from the broken leg by setting him up with a private surgery in London - popped in to offer his drumming skills on the psychedelic ballad Sunday Rain. Justin Timberlake makes a surprise appearance too, providing backing vocals for the Led Zeppelin-style boogie rocker Make It Right. And the producer is Greg Kurstin, a jazz-trained musician who co-wrote Adele's Hello.

It's unashamedly a rock album, but after the stripped-back garage of Wasting Light (2011) and the conceptual contrivance of Sonic Highways (2014), on which each song represented a different city, it's also accessible, tuneful and fun.

"Dave describes it as Motorhead meets Sgt Pepper," Hawkins says. "We took care of the Motorhead part - that's what we do - and Greg came in with these harmonies and some pretty strange jazz arrangements. Honestly, I didn't think it would work. But it did."

Then there is the challenge of reconciling home life with the circus of being in a touring band. "Home is the hard part," Grohl says, sagely.

"Out here we have three things to do: eat, sleep and rock," says Hawkins, who, like Grohl, is married and has three young children. "I can lead a selfish life on the road, eating whatever and whenever I want. Then I have to go home and be a parent."

That leads to the thorny issue of rock stars and parental advice. Grohl's 11-year-old daughter, Violet, decided recently to pick up the flute, only to discover that the instrument is hard enough to put together, let alone play. On receiving an email from her music teacher that Violet wasn't trying, Grohl's wife, Jordyn, told her husband: "You're the musician, you speak to her."

"I gave Violet three options," Grohl says. "One: I can sit with you and figure it out. Two: don't play the fucking flute. Or three: don't be a fucking musician. And then she goes to school the next day and tells her music teacher that her dad said it's OK if she doesn't play the flute any more."

I can't do anything else and I don't want to go back to working in a pizza restaurant

Unfortunately Violet also thinks it's OK not to go to college because her father dropped out of school at 17 to tour with the hardcore punk band Scream. "So I have to tell her, 'You are a completely different person to me and you shouldn't rule out the luxury of learning.' My mother was a teacher, and she realised I was never going to engage in school because I was going to put all my passion, energy and drive into being in a band."

"When I was ten I was a dork: bad at sports, shitty at school, no aptitude for anything," Hawkins says. "Then I sat on a drum kit at my friend's house, played a basic rock beat, and that was that. Now I'll take my mountain bike out on the road in California and see campus life and think, 'Did we miss out?' "

"There was a school fundraiser recently and the theme was 'college'," Grohl says by way of an answer. "I had no idea what to wear and my wife said, 'Don't worry about it, just wear what you would have worn back then.' So I put on high-tops, thermals, shorts, a leather jacket and a baseball hat on backwards - exactly what I wore in Nirvana. And everyone else was in flannel."

A formal education may not be Foo Fighters' strong point, but being in a successful band forces you to learn diplomacy, and at least some books will be consumed in those interminable hours on the tour bus. Grohl wrote The Sky is a Neighborhood, a single from Concrete and Gold, after reading the astrophysicist Neil deGrasse Tyson's theories on how the atoms comprising life on earth are the same atoms scattered through the universe. "That got me thinking of responsibility, not just to each other, but to the planet itself," Grohl says. "Whether it comes to global warming or general good nature, we have to keep our cool."

A lot of people do seem to be losing their cool at the moment. "America is so fucking confused right now," Grohl says. "There is a loss of compassion and decency because the obsession with fame, money and status is out of control and everything has become divided. I'll watch Fox and then CNN and realise they are two different worlds reporting on the same one and it makes no sense."

"You know you're fucked when you miss George Bush Jr, or at least someone presidential," Hawkins says. "And I'm amazed at what is seen as important. I'll click on Yahoo and it's Hurricane Irma, North Korea, the Kardashians? They are the destruction of any culture we ever had in America."

They may not make dad-rock, but the two members of Foo Fighters do sound quite a lot like a pair of hungover, middle-aged veterans moaning about the state of the world. Still, Grohl didn't get to be known as the nicest guy in music for nothing and, as the interview winds up, his inclusive, eminently sensible approach also sounds like a tonic for turbulent times.

"I grew up outside Washington DC and my father was a Republican speechwriter," Grohl says. "Most of his friends were staunch conservatives who I got along great with, even though I was a radical leftist punk rocker. It used to be that no matter which side you sat on the aisle, on Sunday afternoon you could share a whiskey with anyone. Now there is no middle ground. People have the freedom to believe what they want to, but it needs to be based on the human condition of common decency."

If a bunch of high-school dropouts turned stadium rockers can make their community work, perhaps there's hope for the rest of us. "There is a saying in the American South," Grohl says. "At dinner the three things you don't talk about are religion, money and politics. Throw those things out of the window and we can all get along. I think."