Even though the local density approximation leads to an overestimation of the absolute value of the nuclear level density parameter, we show that this simplifying approximation accurately describes the temperature dependence of Thomas-Fermi calculations of the nuclear level density parameter performed by Shlomo and Natowitz. A simple expression that adequately reproduces the Thomas-Fermi calculations of Shlomo and Natowitz is obtained. Two different definitions of the temperature dependent level density parameter can be adopted. Care must be taken to ensure that level density parameters extracted from experimental data are compared to theoretical calculations made using the appropriate definition.