Test of the Einstein equivalence principle with spectral distortions in the cosmic microwave background

The Einstein Equivalence Principle~(EEP) can be verified by the measurement of the spectral distortions of the Cosmic Microwave Background (CMB). The existence of energy-dependency in the cosmological redshift effect means the EEP violation. Introducing the energy-dependent Friedmann-Robertson-Walker metric motivated by rainbow gravity, we show that the energy-dependent redshift effect causes the CMB spectral distortions. Assuming the simple energy-dependent form of the metric, we evaluate the distortions. From the COBE/FIRAS bound, we find that the deviation degree from the EEP, which is comparable to the difference of the parameterized-post-Newtonian parameter "gamma" in energy, is less than 10^{-9} at the CMB energy scale.