Provided is a slewing drive apparatus for a construction machine capable of satisfactory pump-flow-rate control regardless of change in engine speed, including a variable displacement hydraulic pump (10), a slewing motor (12), a slewing operation device (13), a control valve (14), a relief valve (20), and a pump-flow-rate control device (15, 16) that performs a relief cut control and includes: a section for detecting an engine revolution number Ne and a slewing speed of a slewing body (2, 17); a section for determining a target pump flow rate Qo that is a sum of a slewing-speed correspondence flow rate Q1 and a minimum required relief flow rate Qmin; a section for determining a target pump-tilt-angle qtg obtained by dividing the target pump flow rate Qo by the detected engine revolution number Ne; and a section for adjusting an actual pump-tilt-angle of the variable displacement hydraulic pump (10) so as to bring the actual pump-tilt-angle to the target pump-tilt-angle qtg.