  We explore the thermodynamics of a black-hole solution in improved
holographic QCD with a simple dilaton potential having two parameters. By
applying techniques of singular perturbation theory, we get uniform
approximations for the metric and the dilaton field in the two regimes of big
and small black-holes. These techniques lead to a resummation of the naive
expansion at high temperatures, providing an important theoretical improvement
with respect to previous results in the literature. By using this technique, it
is shown how a quadratic dependence at low enough temperatures can naturally
appear in the free energy. A comparison with lattice data of gluodynamics is
performed. It is provided as well an estimate of the value of the gluon
condensate at zero temperature which turns out to be in quite good agreement
with the accepted values in the literature from phenomenological studies of
QCD.
