  This short note considers, within the external field approach outlined in
hep-ph/0202026, the role of the lowest lying gluon Landau mode in QCD in the
high temperature limit. Its influence on a temperature- and field-dependent
running coupling constant is examined. The thermal imaginary part of the mode
is temperature-independent in our approach and exactly cancels the well-known
zero temperature imaginary part, thus rendering the Savvidy vacuum stable.
Combining the real part of the mode with the contributions from the higher
lying Landau modes and the vacuum contribution, a field-independent coupling
alpha_s(T) is obtained. It can be interpreted as the ordinary zero temperature
running coupling constant with average thermal momenta <k> \approx 2pi T for
gluons and <k> \approx pi T for quarks.
