Correctly, it’s spelled OneRepublic. All together with the O and R capitalized. The quickest, and simplest, way to explain why it’s spelled like this is unity; The unity of the fans, the unity of the band, and the way their music is united with each listener. Although, the band doesn’t get upset if you spell it wrong. I mean, their own record label has it spelled wrong on their website.
It mainly comes down to the fans getting more upset if you spell it wrong than anything. As terrible as this sounds, it has almost become a way of “weeding out” the new or casual fans from the older or more dedicated ones.