  There are several theoretical indications that the quantum gravity approaches
may have predictions for a minimal measurable length, and a maximal observable
momentum and throughout a generalization for Heisenberg uncertainty principle.
The generalized uncertainty principle (GUP) is based on a momentum-dependent
modification in the standard dispersion relation which is conjectured to
violate the principle of Lorentz invariance. From the resulting Hamiltonian,
the velocity and time of flight of relativistic distant particles at Planck
energy can be derived. A first comparison is made with recent observations for
Hubble parameter in redshift-dependence in early-type galaxies. We find that
LIV has two types of contributions to the time of flight delay $\Delta t$
comparable with that observations. Although the wrong OPERA measurement on
faster-than-light muon neutrino anomaly, $\Delta t$, and the relative change in
the speed of muon neutrino $\Delta v$ in dependence on redshift $z$ turn to be
wrong, we utilize its main features to estimate $\Delta v$. Accordingly, the
results could not be interpreted as LIV. A third comparison is made with the
ultra high-energy cosmic rays (UHECR). It is found that an essential ingredient
of the approach combining string theory, loop quantum gravity, black hole
physics and doubly spacial relativity and the one assuming a perturbative
departure from exact Lorentz invariance. Fixing the sensitivity factor and its
energy dependence are essential inputs for a reliable confronting of our
calculations to UHECR. The sensitivity factor is related to the special time of
flight delay and the time structure of the signal. Furthermore, the upper and
lower bounds to the parameter, $\alpha$ that characterizes the generalized
uncertainly principle, have to be fixed in related physical systems such as the
gamma rays bursts.
