In extreme temperature environments, a newly emerging engineering application involves both the active and passive control of structures using cryogenic shape memory alloys, which are smart materials able to recover high deformation below the freezing point. With the objective of carrying out new advances in this area, the present work aims to investigate the Cu-7.5Al-13.5Mn (wt.%) shape memory alloy. Thermal, microstructural, and thermomechanical analyses of as-cast and hot-rolled specimens were performed, taking into account the effects of annealing and solubilization. It was observed that the phase transition occurs at temperatures below 120 K and changes according to the thermo-mechanical path. Specifically, hot-rolling lowers the phase transition temperature range with respect to the as-cast condition–from 34 K to 23 K for Mf, and from 89 K to 80 K for Af. Additionally, when the annealing temperature rises, the phase transformation temperature increases as well, and the alloy loses its cryogenic features when heat treated above 473 K. Finally, loss factors of 0.06 and 0.088, which were respectively found in dynamic and static settings, validate the material’s good damping response.