•We analyse the data of the energetic photons from the gamma-ray bursts (GRBs).•We find events to support the energy dependence in the light speed.•We provide scenarios to understand the Fermi-LAT Pass 8 data of bright GRBs.•The predictions from such scenarios are testable by future detected GRBs. The constancy of light speed is a basic assumption in Einstein’s special relativity, and consequently the Lorentz invariance is a fundamental symmetry of space–time in modern physics. However, it is speculated that the speed of light becomes energy-dependent due to the Lorentz invariance violation (LV) in various new physics theories. We analyse the data of the energetic photons from the gamma-ray bursts (GRBs) by the Fermi Gamma-Ray Space Telescope, and find more events to support the energy dependence in the light speed with both linear and quadratic form corrections. We provide two scenarios to understand all the new-released Pass 8 data of bright GRBs by the Fermi-LAT Collaboration, with predictions from such scenarios being testable by future detected GRBs.