Apparatus for the forming of a strip, formable at increased temperature, in particular of a strip consisting of thermoplastic, in a continuous run on the surface of a rotating drum which is to be heated and/or to be cooled in predetermined stationary regions. In order to reduce the energy losses due to the heating and cooling alternation, the thin-walled drum is mounted on a carrier body which can be heated or cooled in the respective regions and is heat-exchangeable with the drum surface at least in these regions.