By applying linear-response methods long familiar in many-body theory to quantum fluctuations in curved spacetime, the vacuum is shown to behave like a dissipative as well as a fluctuating medium. The implications of this quantum vacuum ''viscosity'' for cosmology and the cosmological constant are discussed. Since the de Sitter phase is unstable to particle creation processes, this vacuum dissipative mechanism may be responsible for the approximate flatness of the observed Universe.