A system includes a variable displacement pump (VDP) in fluid communication with an inlet line and with an outlet line. The VDP includes a variable displacement mechanism configured to vary pressure to the outlet line. A bypass valve (BPV) includes a BPV inlet in fluid communication with the outlet line, and a BPV outlet in fluid communication with a bypass line that feeds into the inlet line upstream of the VDP. The BPV includes a thermostatic actuator configured to control recirculation flow amount through the BPV based on fluid temperature in the BPV. A method includes thermostatically controlling a bypass valve (BPV) to recirculate flow from the outlet line to an input line of the VDP in the event of flow demanded by the downstream system dropping below a predetermined low threshold of flow through the VDP based on fluid temperature in the BPV.