Answers to: How have space suits dissipated the heat removed from astronauts? explain how USA suits currently use open-loop cooling.
Typically, space suits must dissipate several hundred watts of heat from the body of an EVAing astronaut.
The Apollo and Shuttle/ISS EVA suits used open-loop cooling by the sublimation of water into (near) vacuum, which introduced an additional consumable and is unworkable in an atmosphere.
Have any spacesuits been designed that use radiators (and presumably heat pumps) to dissipate heat? My calculations suggest that with a radiator temperature of ~ 500 K (227 deg C) you can get around 3.5 kW per square meter, which seems manageable.