  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 membrane's 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$_2$.
