Time delay of light signals in an energy-dependent spacetime metric
  In this note we review the problem of time delay of photons propagating in a spacetime with a metric that explicitly depends on the energy of the particles (Gravity-Rainbow approach). We show that corrections due to this approach -- which is closely related to DSR proposal -- produce for small redshifts ($z<<1$) smaller time delays than in the generic Lorentz Invariance Violating case. 
A. F. Grillo, E. Luzio and F. Mendez