  Familiar concepts in physics, such as Lorentz symmetry, are expected to be
broken at energies approaching the Planck energy scale as predicted by several
quantum-gravity theories. However, such very large energies are unreachable by
current experiments on Earth. Current and future Cherenkov telescope facilities
may have the capability to measure the accumulated deformation from Lorentz
symmetry for photons traveling over large distances via energy-dependent time
delays. One of the best natural laboratories to test Lorentz Invariance
Violation~(LIV) signatures are Gamma-ray bursts~(GRBs). The calculation of time
delays due to the LIV effect depends on the cosmic expansion history. In most
of the previous works calculating time lags due to the LIV effect, the standard
$\Lambda$CDM (or concordance) cosmological model is assumed. In this paper, we
investigate whether the LIV signature is significantly different when assuming
alternatives to the $\Lambda$CDM cosmological model. Specifically, we consider
cosmological models with a non-trivial dark-energy equation of state ($w \neq
-1$), such as the standard Chevallier-Polarski-Linder~(CPL) parameterization,
the quadratic parameterization of the dark-energy equation of state, and the
Pade parameterizations. We find that the relative difference in the predicted
time lags is small, of the order of at most a few percent, and thus likely
smaller than the systematic errors of possible measurements currently or in the
near future.
