A cooling system for an engine is disclosed. The cooling system may have a pump driven by the engine to pressurize coolant, a single heat exchanger configured to cool pressurized coolant, and an aftercooler configured to transfer heat from air entering the engine to pressurized coolant. The cooling system may also have a multi-thermostat arrangement configured to always direct at least a portion of the pressurized coolant from the pump through the aftercooler, and selectively direct pressurized coolant from the pump through the single heat exchanger. The multi-thermostat arrangement may also be configured to selectively mix pressurized coolant from the aftercooler with a remaining portion of the pressurized coolant from the pump to form a coolant mixture, and to selectively direct the coolant mixture through the engine or around the engine.