Well I can't speak strictly for or against any of this, but I do have an anecdotal story for you. I had a friend in undergrad who was a math/physics double major. He did nearly all of his research in physics. The research he did included 2 REUs* in physics, some research during the course of 2 school years with two different professors doing physics work including an honors thesis in physics. Technically one of the physics projects he worked on was with a math professor in the math department. She was doing research in applied math simulations and doing physical experiments (personally I'd call that physics but maybe that's just my bias). He ended up going to grad school in math (math math, not applied math). He went to a good math program, fully funded, etc.
Point of this story: this friend clearly focused a lot on physics (and technically applied math) research and did just fine in applications. All of the work he did was actually in simulations, but it was all applied to physics problems, even the applied mathematician he worked with was really doing physics experiments and simulations. I don't think it makes THAT big a difference, particularly if you're double majoring in math and physics. Considering you want to do theoretical physics, math research isn't that big a leap away.
All that being said, I'd agree with kroner: physics research is probably more beneficial on applications than math research. There's always the suggestion that maybe doing math research might make your application more interesting as most of the applicants would be doing physics research?
P.S. *The REUs he did weren't technically REUs; they were just funded summer internships.