The question of whether unobserved short-wavelength modes of the gravitational field can induce decoherence in the long-wavelength modes (``the decoherence of spacetime'') is addressed using a simplified model of perturbative general relativity, related to the Nordstrom-Einstein-Fokker theory, where the metric is assumed to be conformally flat. For some long-wavelength coarse grainings, the Feynman-Vernon influence phase is found to be effective at suppressing the off-diagonal elements of the decoherence functional. The requirement that the short-wavelength modes be in a sufficiently high-temperature state places limits on the applicability of this perturbative approach.