The non-equilibrium non-Markovian barrier crossing dynamics of a one-dimensional massive coordinate, described by the non-equilibrium version of the generalized Langevin equation with unequal random and friction relaxation times, is studied by simulations and analytical methods. Within a harmonic approximation, a general formula for the barrier crossing time is derived which agrees favorably with simulations. Non-equilibrium random forces with a relaxation time longer than the friction relaxation time induce non-Arrhenius behavior and dramatically increase the barrier crossing time; within the harmonic theory this corresponds to a reduced effective temperature which also modifies the spatial and velocity distributions.
