If your thermostat is newly installed and the heat pump worked for cooling with the previous thermostat, check the following:
1. Verify that the thermostat is set to cool and is calling for cooling, usually indicated by the words "cool on," a snowflake icon, or a shimmering snowflake icon. If the room temperature is below the cool set point, the thermostat display should indicate a call for cooling
2. Confirm that the circuit breakers are on.
3. If you installed the thermostat yourself, ensure that the wires are matched correctly and securely connected at the thermostat. Also verify that the thermostat is correctly configured to control a heat pump. The installation instructions for your thermostat show these settings.
4. If the heat pump runs in heat mode with a call for cooling, verify that the configuration setting for the changeover is set correctly for your system. The configuration settings are shown in the installation guide for your thermostat. If you do not know whether your heat pump uses the changeover valve in heat or cool mode, call your professional HVAC contractor or the manufacturer of your heat pump.
5. If there are not separate wire connections to R and Rc on the thermostat, those terminals need a jumper. Some models use a small metal jumper and other models have a switch that needs to be set to "R".
If the same issue occurred with the previous thermostat, then the problem is most likely with your heat pump. Call a professional HVAC contractor if the above steps don't solve the issue.