We present detailed measurements of the differential resistance (dV/dI) of state-of-the-art FM/AlOx/FM magnetic tunnel junctions (MTJ) as a function of applied bias and temperature. Temperature effects are particularly significant in physical quantities involving narrow features such as those at low-voltage bias. We show that the temperature evolution of the tunneling characteristics and, in particular, the pronounced rounding of the dV/dI curves with increasing temperature can be well explained by thermal smearing of the tunneling electron energy distribution.