The PHENIX collaboration presents first measurements of low-momentum (0.4 < p_T < 3 GeV = c) direct-photon yields from Au + Au collisions at √^s_NN = 39 and 62.4 GeV. For both beam energies the direct-photon yields are substantially enhanced with respect to expectations from prompt processes, similar to the yields observed in Au + Au collisions at √^s_NN = 200. Analyzing the photon yield as a function of the experimental observable dN_ch /dη reveals that the low-momentum (> 1 GeV/c) direct-photon yield dN^dir_γ /dη is a smooth function of dN_ch /dη and can be well described as proportional to (dN_ch /dη)^α with α ≈ 1.25. This scaling behavior holds for a wide range of beam energies at the Relativistic Heavy Ion Collider and the Large Hadron Collider, for centrality selected samples, as well as for different A + A collision systems. At a given beam energy, the scaling also holds for high p_T (> 5 GeV/c), but when results from different collision energies are compared, an additional √^s_NN -dependent multiplicative factor is needed to describe the integrated-direct-photon yield.