The medium-temperature $T$ dependence of the jet transport coefficient $\hat q$ was studied via the nuclear modification factor $R_{AA}(p_{\rm T})$ and elliptical flow parameter $v_2(p_{\rm T})$ for large transverse momentum $p_{\rm T}$ hadrons in high-energy nucleus-nucleus collisions. Within a next-to-leading-order perturbative QCD parton model for hard scatterings with modified fragmentation functions due to jet quenching controlled by $\hat q$, we check the suppression and azimuthal anisotropy for large $p_{\rm T}$ hadrons, and extract $\hat q$ by global fits to $R_{AA}(p_{\rm T})$ and $v_2(p_{\rm T})$ data in A + A collisions at RHIC and LHC, respectively. The numerical results from the best fits show that $\hat q/T^3$ goes down with local medium temperature $T$ in the parton jet trajectory. Compared with the case of a constant $\hat{q}/T^3$, the going-down $T$ dependence of $\hat{q}/T^3$ makes a hard parton jet to lose more energy near $T_{\rm c}$ and therefore strengthens the azimuthal anisotropy for large $p_{\rm T}$ hadrons. As a result, $v_2(p_{\rm T})$ for large $p_{\rm T}$ hadrons was enhanced by approximately 10% to better fit the data at RHIC/LHC. Considering the first-order phase transition from QGP to the hadron phase and the additional energy loss in the hadron phase, $v_2(p_{\rm T})$ is again enhanced by 5%-10% at RHIC/LHC.
