It is shown using general arguments that at any location along an optical fiber laser or amplifier, the temperature variation across the fiber is two to three orders of magnitude smaller than the difference between the fiber surface temperature and the ambient temperature. Therefore, the fiber heats almost uniformly in its transverse cross-section regardless of how hot it gets relative to its surroundings, and regardless of how the temperature varies along the fiber. The analysis applies to a broad range of fiber laser and amplifier designs subject to convective air-cooling but can be readily adapted to other forms of cooling.