The apparatus comprises an internal and/or external multiple-stage cooling device (40, 47) arranged coaxially with a drawing orifice (H) of an axtruder die (E), and having multi-level tangential outlets for the coolant to stabilize a non-stabilized section (M) of the expanded foil hose (F) by spiral coolant streams (46, 48). The internal device (40) comprises cooling units (41-44) arranged at axial distance from each other, surrounding the foil section (M) through a gap (G1). Each cooling unit (41-44) is connected to a coolant supply (45) to supply the coolant of selectively adjustable temperature.The external cooling device (47) comprises at least two cooling units (47.1, 47.2, 47.3) arranged at axial distance from each other, surrounding the foil section (M) through a gap (G). Each cooling unit (41-44) is provided with tangential inlets (17, 18) and is connected to a coolant supply (60) to supply the coolant of selectively and individually adjustable temperature and/or volume and/or pressure.