  We apply the previously proposed scheme of approximately self-consistent
hard-thermal-loop resummations in the entropy of high-temperature QCD to N=4
supersymmetric Yang-Mills (SYM) theories and compare with a (uniquely
determined) R[4,4] Pad\'e approximant that interpolates accurately between the
known perturbative result and the next-to-leading order strong-coupling result
obtained from AdS/CFT correspondence. We find good agreement up to couplings
where the entropy has dropped to about 85% of the Stefan-Boltzmann value. This
is precisely the regime which in purely gluonic QCD corresponds to temperatures
above 2.5 times the deconfinement temperature and for which this method of
hard-thermal-loop resummation has given similar good agreement with lattice QCD
results. This suggests that in this regime the entropy of both QCD and N=4 SYM
is dominated by effectively weakly coupled hard-thermal-loop quasiparticle
degrees of freedom. In N=4 SYM, strong-coupling contributions to the
thermodynamic potential take over when the entropy drops below 85% of the
Stefan-Boltzmann value.
