Glass manufacturing apparatuses with cooling devices and methods for using the same are disclosed. In one embodiment, an apparatus for forming a glass web from molten glass includes an enclosure and pulling rolls that cooperate to draw a glass web in a draw direction rotatably positioned in an interior of the enclosure. A cooling device for extracting heat from the glass web is in fluid communication with a cooling fluid source and includes an actively cooled flapper disposed in the interior of the enclosure that is movable to facilitate varying the heat extraction. The actively cooled flapper serves as a heat sink in the interior of the enclosure and the cooling fluid extracts heat from the actively cooled flapper to remove heat from the glass web and the enclosure.