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.