We derive a power-optimizing output feedback controller for a multi-zone heat pump that (1) regulates individual zone temperatures, rejecting unknown heat load disturbances, (2) regulates condenser subcooling and (3) the compressor discharge temperature, and (4) minimizes electrical power consumption at steady-state operating conditions. The design is a cascade of a linear inner-loop and a nonlinear outer-loop. The inner-loop is designed for robust disturbance rejection using H-infinity loop-shaping methods. The outer-loop uses a model of compressor and fan power consumption and a gradient descent feedback to drive the system to its powerminimizing equilibrium for constant values of references and disturbances. The controller uses only temperature measurements for feedback; refrigerant pressure sensors, which are not present in many products for cost reasons, are not required. A proof of exponential stability is provided and preliminary experimental tests demonstrate satisfactory transient responses for a commercial multi-zone heat pump.