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The Stefan-Boltzmann law governs the temperature dependence of the blackbody emission of radiation: . A consequence of this nonlinearity is that a cold object needs a greater increase in temperature than a hot object in order to reach the same increase in radiation emitted. Therefore, this nonlinearity potentially has an impact on the structure of radiatively forced atmospheric temperature change in both the horizontal and vertical directions. For example, it has previously been argued to be a cause of polar amplification (PA) of surface air warming. Here, the role of this nonlinearity is investigated by 1) assessing the magnitude of its effect on PA compared to spatial variations in CO2's radiative forcing for Earth's atmosphere and 2) linearizing in a gray radiation atmospheric general circulation model (GCM) with an interactive hydrological cycle. Estimates for Earth's atmosphere show that the combination of the Planck feedback and forcing from CO2 would produce a tropically amplified warming if they were the only means of changing the Earth's energy balance. Contrary to expectations, climate change simulations with linearized radiation do not have reduced polar amplification of surface air warming relative to the standard GCM configuration. However, simulations with linearized radiation consistently show less warming in the upper troposphere and more warming in the lower troposphere across latitudes. The lapse rate feedbacks from pure radiative and radiative-convective configurations of the model are used to show that the cold-altitudes-warm-more effect of the nonlinearity carries across this model hierarchy.