We measure the thermal time constants of suspended single layer molybdenum disulfide drums by their thermomechanical response to a high-frequency modulated laser. From this measurement the thermal diffusivity of single layer MoS$_2$ is found to be 1.14 $times$ 10$^{-5}$ m$^2$/s on average. Using a model for the thermal time constants and a model assuming continuum heat transport, we extract thermal conductivities at room temperature between 10 to 40 W/(m$cdot$K). Significant device-to-device variation in the thermal diffusivity is observed. Based on statistical analysis we conclude that these variations in thermal diffusivity are caused by microscopic defects that have a large impact on phonon scattering, but do not affect the resonance frequency and damping of the membranes lowest eigenmode. By combining the experimental thermal diffusivity with literature values of the thermal conductivity, a method is presented to determine the specific heat of suspended 2D materials, which is estimated to be 255 $pm$ 104 J/(kg$cdot$K) for single layer MoS$_2$.
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