The c(5√2×√2)R45°-Pb/Cu(100) surface phase is investigated by means of angle resolved ultraviolet photoemission and low energy electron diffraction in the temperature range between 300 and 550K. We identify and characterize a temperature-induced surface phase transition at 440K from the room temperature c(5√2×√2) R45° phase to a (√2×√2)R45° structure with split superstructure spots. The phase transition is fully reversible and takes place before the two-dimensional melting of the structure at 520K. The electronic structure of the split (√2×√2)R45° phase is characterized by a metallic free-electron like surface band. This surface band is backfolded with c(5√2×√2)R45° periodicity phase at room temperature, giving rise to a surface band gap at the Fermi energy. We propose that a gain in electronic energy explains in part the stability of the c(5√2×√2)R45° phase.