This paper proposes a shape memory alloy (SMA)-based artificial muscle that has an active cooling system. An SMA coil spring is embedded into a stretchable coolant vessel, which allows active cooling of SMA through coolant circulation. The coolant effect has been investigated by measuring actuation speed. Integrating a thermomechanical model and heat transfer model between coolant and SMA, actuation times have been estimated. Based on these results, actuation performances that use four different coolants have been found out, and the mineral oil cooling case shows both contraction and extension times of less than 1 second to contract to over 35 % of its initial length with 1 kg load. A soft wearable robot for assisting wrist motion has been designed by integrating multiple SMA muscles with a glove, wristband, and elbow anchor. Thanks to the advantages of the SMA coil spring actuator, the proposed robot is compliant and weight-balanced structure. The robot shows ranges of motion (ROM) of 38, 50, 34, and 35, for flexion, extension, radial deviation, and ulnar deviation, respectively. In addition, the robot shows improved cyclic performance compared to a natural air-cooling system. Finally, its torque assistance and the amount of time needed to wear it have been analyzed with user tests.