The temperature dependence of multiphonon nonradiative transitions, caused by linear diagonal and quadratic nondiagonal vibronic interactions, is investigated on the basis of nonperturbative quantum theory. Both transitions up and down in energy are considered. It was found that the usual increase in transition rate with temperature may not hold near some (critical) value(s) of a nondiagonal interaction and temperature, when the rate becomes very high (comparable to mean phonon frequency), and its temperature dependence exhibits a maximum. The results may be important for understanding the mechanisms of catalysis in chemical reactions.