This investigation concerns the energetics of a Cu double layer (DL) on Mo(110) phase transition involving: one-dimensional (1D) row matching, a rotation of the matching direction through 90 degrees, a highly distorted high temperature (HT) substrate stabilized polymorphic phase, comprising two identical (100) Cu monolayers (MLs), and a low temperature (LT) phase, comprising a pseudomorphic tps) ML below and an almost close-packed (cp) ML above, A primary goal here is to show that the zero temperature internal energies of the two phases differ so little that the difference Delta epsilon(HL) could be overcome by temperature dependent free energy contributions. Quantification of the analytical results, using embedded-atom (EAM) potentials and averaging procedures of periodic quantities, confirmed that the difference Delta epsilon(HL) is small. Crude estimates point towards atomic vibration frequency differences as the most likely candidate to generate the T-dependent free energy to overcome the gap Delta epsilon(HL). The investigation otherwise also yielded results of wider interest.