The lifetimes were measured for rovibronic levels lying slightly lower than the top of the potential barrier in the H-2 EF(1)Sigma (+)(g) double-minimum state, with a delayed coincidence method following electron impact excitation from the ground state. Dependence of lifetimes on the rotational levels in a pair of vibrational levels is explained in terms of perturbation caused by rotation-vibration interaction between these levels. Lifetimes of some D(1)Pi (u) and B'(1)Sigma (+)(u) levels were also measured to estimate the cascade effects.