The evolution with temperature of the elastic constants of the metastable β phase in a Cu-27.96 at.%Al-3.62 at.% Ni shape memory alloy (SMA) has been studied by resonant ultrasound spectroscopy (RUS). The corresponding elastic constants of this cubic phase have been measured near the martensitic transformation temperature in a single crystal. Above the martensitic transformation temperature, an anomalous behavior has been found in the C′elastic constant which shows a softening as the temperature decreases. The internal friction value Q−1 has been obtained in this temperature range from the RUS spectra. The mechanisms associated with the Q−1 increase must be related to {1 1 0} shear.