Neutral-pion π(0) spectra were measured at midrapidity (|y|<0.35) in Au+Au collisions at √(s(NN))=39 and 62.4 GeV and compared with earlier measurements at 200 GeV in a transverse-momentum range of 1<p(T)<10 GeV/c. The high-p(T) tail is well described by a power law in all cases, and the powers decrease significantly with decreasing center-of-mass energy. The change of powers is very similar to that observed in the corresponding spectra for p+p collisions. The nuclear modification factors (R(AA)) show significant suppression, with a distinct energy, centrality, and p(T) dependence. Above p(T)=7 GeV/c, R(AA) is similar for √(s(NN))=62.4 and 200 GeV at all centralities. Perturbative-quantum-chromodynamics calculations that describe R(AA) well at 200 GeV fail to describe the 39 GeV data, raising the possibility that, for the same p(T) region, the relative importance of initial-state effects and soft processes increases at lower energies. The p(T) range where π(0) spectra in central Au+Au collisions have the same power as in p+p collisions is ≈5 and 7 GeV/c for √(s(NN))=200 and 62.4 GeV, respectively. For the √(s(NN))=39 GeV data, it is not clear whether such a region is reached, and the x(T) dependence of the x(T)-scaling power-law exponent is very different from that observed in the √(s(NN))=62 and 200 GeV data, providing further evidence that initial-state effects and soft processes mask the in-medium suppression of hard-scattered partons to higher p(T) as the collision energy decreases.