I depends on how the thermostat is joined to the Z-Wave controller.
If the Z-Wave module was joined to the Z-Wave controller with the thermostat only powered by batteries it will not act as a repeater. In this mode the thermostat conserves battery by allowing the module to sleep when not needed. The device wakes up periodically to retrieve commands from the controller.
It the Z-Wave module is joined to the Z-Wave controller with the thermostat powered by a C wire the Z-Wave thermostat will as as a repeater node on the Z-Wave network.