To uniformly cool a high-temperature steel workpiece when passing through this cooling box which is a hollow cooling box for steel mills.

This box has an interior plenum which is formed with members with which the cooling box is formed. A cooling medium is supplied to this plenum under pressure and the cooling medium passes through a curved wedge-shaped slots which are one of members opposing to the workpiece when passing through the cooling box. The wedge-shaped slots are oriented a direction so that the widest part of the slot is on the upstream side of the passage of the workpiece.