  In the paper, we analyze the properties of Gross-Llewellyn Smith (GLS) sum
rule by using the $\mathcal{O}(\alpha_s^4)$-order QCD corrections with the help
of principle of maximum conformality (PMC). By using the PMC single-scale
approach, we obtain an accurate renormalization scale-and-scheme independent
fixed-order pQCD contribution for GLS sum rule, e.g. $S^{\rm GLS}(Q_0^2=3{\rm
GeV}^2)|_{\rm PMC}=2.559^{+0.023}_{-0.024}$, where the error is squared average
of those from $\Delta\alpha_s(M_Z)$, the predicted
$\mathcal{O}(\alpha_s^5)$-order terms predicted by using the Pad\'{e}
approximation approach. After applying the PMC, a more convergent pQCD series
has been obtained, and the contributions from the unknown higher-order terms
are highly suppressed. In combination with the nonperturbative high-twist
contribution, our final prediction of GLS sum rule agrees well with the
experimental data given by the CCFR collaboration.
