In a hydraulic pump control apparatus and method of a construction machine, the control apparatus includes a pump control unit for controlling discharge pressure of a pump driven by an engine. The control unit includes: a pressure setting value calculating unit which calculates a pressure setting value based on an engine output torque estimating value or an engine RPM; and a breakdown treating unit selects one of the pressure setting value and a pressure command value according to a breakdown of the swash plate angle sensor to output the selected value. Since the pump is controlled according to a pressure setting value calculated based on the engine output torque estimating value, the absorption torque value of the pump does not exceed the maximum torque value of the engine even when the swash plate angle sensor breaks down, and related engine stoppage is prevented even during a high-load operation.