Coulomb instability in hot nuclei is discussed, using the Skyrme effective nucleon-nucleon interactions and a mean-field approach based on a finite-temperature real-time Green's-function method. In the calculation, full degeneracy and symmetry corrections are included. It is found that the above precise treatment of degeneracy and symmetry corrections is necessary in the lower-temperature and higher-density region or for larger asymmetry. The calculated limiting temperatures are quite different for different sets of Skyrme effective interactions, which may be used as a sensitive test of the various Skyrme interactions to see which one can best describe the finite-temperature properties of nuclear matter. It is also found that the limiting temperature ${\mathit{T}}_{\mathrm{lim}}$ is approximately proportional to the critical temperature ${\mathit{T}}_{\mathit{C}}$, when the results for various Skyrme interactions are compared.