The mass of the gaseous reservoir in young circumstellar disks is a crucial initial condition for the formation of planetary systems, but estimates vary by orders of magnitude. In some disks with resolvable cavities, sharp inner disk warps cast two-sided shadows on the outer rings; can the cooling of the gas as it crosses the shadows bring constraints on its mass? The finite cooling timescale should result in dust temperature decrements shifted ahead of the optical/IR shadows in the direction of rotation. However, some systems show temperature drops, while others do not. The depth of the drops and the amplitude of the shift depend on the outer disk surface density Sigma through the extent of cooling during the shadow crossing time, and also on the efficiency of radiative diffusion. These phenomena may bear observational counterparts, which we describe with a simple one-dimensional model. An application to the HD142527 disk suggests an asymmetry in its shadows, and predicts a >~10deg shift for a massive gaseous disk, with peak Sigma > 8.3 g/cm2. Another application to the DoAr44 disk limits the peak surface density to Sigma < 13g/cm2
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