"I spoke to PF Sloan himself on the phone the other night, and he said, 'I'm praying for you.'" It's three songs into Rumer's set, and the audience titter uncertainly. Who is PF Sloan? Does Rumer warrant prayer? The answer to the first: he's a revered American songwriter who is the subject of one of tonight's songs. As for the second, only Rumer can say. Since the surprise success of her 2010 debut, Seasons of My Soul, she has been hailed as a progenitor of a credible new wave of MOR, winning awards and playing the White House. But she was also diagnosed as bipolar and recently told the Guardian: "I'm trying very hard not to be depressed, even though I am."

Tonight she's a smiling presence in black lace, standing in a square of evening sunshine. Dreamily fixing her gaze on the ceiling, she lets her golden voice do the work of finding each song's mainspring. The track PF Sloan pours out, sweetly and sincerely: "I have been seeking PF Sloan, but no one knows where he has gone." Next is Townes van Zandt's Flyin' Shoes, a song about escape, and it feels as though running away has crossed her mind.

Flanked protectively by a nine-piece band, she's here to introduce her new album, Boys Don't Cry. It's not the official follow-up to Seasons, which she's still working on, she tells us. Instead, it's a collection of covers of 1970s songs by male artists, and if that weren't sufficiently niche, most are obscure enough for her to tentatively inquire: "Do you know any of these?"

Rumer only discovered most of the songs recently, she says – imagine a male singer admitting that – and, to be honest, they're a mixed bag. Her smooth treatment turns Ronnie Lane's Just for a Moment and Todd Rundgren's Be Nice to Me into cruise-ship fare, but a stark version of Neil Young's A Man Needs a Maid is glorious. So is the encore of her hit Slow: "A precious song," she calls it, to roars of agreement.