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On gravity's rainbow, the energy of particles affects the geometry of blackhole in such a way that the Hawking temperature of black hole is appropriatelymodified. It means that the fiducial and free-fall temperatures on thebackground of black hole should also be modified according to deformation ofthe geometry. We find that the fiducial temperature takes the blue-shiftedmodified Hawking temperature so that it is divergent at the horizon, while thefree-fall temperature defined by employing the thermodynamic Stefan-Boltzmannrelation on gravity's rainbow is finite everywhere without the blue-shifteffect; in particular, it is vanishing at the horizon. In this respect, thebehaviors of the two-different temperatures at the horizon show that black holecomplementarity is still required on gravity's rainbow.

Hawking, Fiducial, and Free-fall Temperatures of Black Hole on Gravity's Rainbow