An apparatus for hot stamping of metal sheets, comprising: - a first die portion; - a second die portion, which defines, upon coupling with the first die portion, a die cavity configured to impart a desired shape on a metal sheet that is to undergo stamping, the die cavity being delimited by walls (2W; 16W) of the first die portion and walls (10W, 12W; 16W) of the second die portion; - a first cooling jacket provided in at least part of the first die portion, the first cooling jacket being configured to operate in a relation of heat exchange with the die cavity through at least part of the walls (2W, 16W) of the first die portion that define the die cavity; and - a second cooling jacket provided in at least part (10, 12) of the second die portion, the second cooling jacket being configured to operate in a relation of heat exchange with the die cavity through at least part of the walls (10W, 12W, 16W) of the second die portion that define the die cavity, wherein at least one of said first die portion and second die portion comprises one or more mobile elements having an outer surface defining a portion of wall that delimits the die cavity, each mobile element being operable to vary the relative position of said outer surface and of the walls (2W, 10W, 12W; 16W) that delimit the die cavity, the apparatus being characterized in that said one or more mobile elements are operable by an outwards movement so that the outer surface thereof projects into the die cavity with respect to the walls of the die cavity adjacent to the mobile element concerned in order to increase the quenching rate, and in that the one or more mobile elements are operable by a retraction so that the outer surface thereof is brought back towards the outside of the die cavity with respect to the walls of the die cavity adjacent to the mobile element involved in order to reduce the quenching rate. - a first die portion; - a second die portion, which defines, upon coupling with the first die portion, a die cavity configured to impart a desired shape on a metal sheet that is to undergo stamping, the die cavity being delimited by walls (2W; 16W) of the first die portion and walls (10W, 12W; 16W) of the second die portion; - a first cooling jacket provided in at least part of the first die portion, the first cooling jacket being configured to operate in a relation of heat exchange with the die cavity through at least part of the walls (2W, 16W) of the first die portion that define the die cavity; and - a second cooling jacket provided in at least part (10, 12) of the second die portion, the second cooling jacket being configured to operate in a relation of heat exchange with the die cavity through at least part of the walls (10W, 12W, 16W) of the second die portion that define the die cavity, wherein at least one of said first die portion and second die portion comprises one or more mobile elements having an outer surface defining a portion of wall that delimits the die cavity, each mobile element being operable to vary the relative position of said outer surface and of the walls (2W, 10W, 12W; 16W) that delimit the die cavity, the apparatus being characterized in that said one or more mobile elements are operable by an outwards movement so that the outer surface thereof projects into the die cavity with respect to the walls of the die cavity adjacent to the mobile element concerned in order to increase the quenching rate, and in that the one or more mobile elements are operable by a retraction so that the outer surface thereof is brought back towards the outside of the die cavity with respect to the walls of the die cavity adjacent to the mobile element involved in order to reduce the quenching rate.