In order to meet the requirements of the Southern Advanced Photon Source (SAPS), a slot-coupled π-mode 5-cell copper cavity with higher order mode (HOM) damping was proposed and developed. In this study, we present the optimal design of a 5-cell copper cavity and an investigation of HOM impedances. A prototype of 5-cell copper cavity based on this design was manufactured. The RF characteristics of 5-cell copper cavity have been tested. The results show that a flat accelerating electric field is obtained, with a coupling coefficient of 1.7% and a shunt impedance more than 19.5 MΩ/m. To damp these HOMs with high impedance, two inductive HOM couplers and two capacitive HOM couplers were designed. Finally, the impedances of all the HOMs are below the threshold of coupled-bunch instabilities (CBI) for SAPS.