  The synchrotron external shock model predicts the evolution of the spectral
($\beta$) and temporal ($\alpha$) indices during the gamma-ray burst (GRB)
afterglow for different environmental density profiles, electron spectral
indices, electron cooling regimes, and regions of the spectrum. We study the
relationship between $\alpha$ and $\beta$, the so-called "closure relations"
with GRBs detected by \textit{Fermi} Large Area Telescope (\textit{Fermi}-LAT)
from 2008 August to 2018 August. The spectral and temporal indices for the >
100 MeV emission from the \textit{Fermi}-LAT as determined in the Second
Fermi-LAT Gamma-ray Burst Catalog (2FLGC) are used in this work. We select GRBs
whose spectral and temporal indices are well constrained (58 long-duration GRBs
and 1 short-duration GRBs) and classify each GRB into the best-matched
relation. As a result, we found that a number of GRBs require a very small
fraction of the total energy density contained in the magnetic field
($\epsilon_{B}$ $\lesssim$ 10$^{-7}$). The estimated mean and standard
deviation of electron spectral index $\mathit{p}$ are 2.40 and 0.44,
respectively. The GRBs satisfying a closure relation of the slow cooling tend
to have a softer $\mathit{p}$ value compared to those of the fast cooling.
Moreover, the Kolmogorov--Smirnov test of the two $\mathit{p}$ distributions
from the fast and slow coolings rejects a hypothesis that the two distributions
are drawn from the single reference distribution with a significance of 3.2
$\sigma$. Lastly, the uniform density medium is preferred over the medium that
decreases like the inverse of distance squared for long-duration GRBs.
