We study diffusion of heat in an aqueous suspension of disc shaped nanoparticles of Laponite, which has finite elasticity and paste-like consistency, by using the Mach-Zehnder interferometer. We estimate the thermal diffusivity of the suspension by comparing the experimentally obtained temperature distribution to that with analytical solution. We observe that despite highly constrained Brownian diffusivity of particles owing to its soft glassy nature, suspensions at very small concentrations of Laponite demonstrates significant enhancement in thermal diffusivity. We correlate the observed enhancement with the possible microstructures of the Laponite suspension.