We perform molecular dynamics simulations to study shape changes of carbon fullerenes and nanotubes with increasing temperature. At moderate temperatures, these systems gain structural and vibrational entropy by exploring the configurational space at little energy cost. We find that the soft phonon modes, which couple most strongly to the shape, maintain the surface area of these hollow nanostructures. In nanotubes, the gain in entropy translates into a longitudinal contraction, which reaches a maximum at T approximately 800 K. Only at much higher temperatures do the anharmonicities in the vibration modes cause an overall expansion.