  We extend the previously obtained results for the thermodynamic potential of
hot QCD in the limit of large number of fermions to non-vanishing chemical
potential. We give exact results for the thermal pressure in the entire range
of temperature and chemical potential for which the presence of a Landau pole
is negligible numerically. In addition we compute linear and non-linear quark
susceptibilities at zero chemical potential, and the entropy at small
temperatures. We compare with the available perturbative results and determine
their range of applicability. Our numerical accuracy is sufficiently high to
check and verify existing results, including the recent perturbative results by
Vuorinen on quark number susceptibilities and the older results by Freedman and
McLerran on the pressure at zero temperature and high chemical potential. We
also obtain a number of perturbative coefficients at sixth order in the
coupling that have not yet been calculated analytically. In the case of both
non-zero temperature and non-zero chemical potential, we investigate the range
of validity of a scaling behaviour noticed recently in lattice calculations by
Fodor, Katz, and Szabo at moderately large chemical potential and find that it
breaks down rather abruptly at $\mu_q \gtrsim \pi T$, which points to a
presumably generic obstruction for extrapolating data from small to large
chemical potential. At sufficiently small temperatures $T \ll \mu_q$, we find
dominating non-Fermi-liquid contributions to the interaction part of the
entropy, which exhibits strong nonlinearity in the temperature and an excess
over the free-theory value.
