We report the thermal conductivity, kappa, and the specific heat, C-p, of dispersion of 0.95-nm-thick, 25-nm-diameter disks of Laponite in water and in ice, as well as the thermal effects during gelation of several compositions, and the temperature dependence of the gelation time. The kappa values of its 5.0 wt% sol and gel states at T>273 K are similar to 3 % larger than those of pure water. In the frozen state of water, kappa is lower than that of hexagonal ice and the difference increases on cooling. kappa of the sol and gel calculated from Maxwell's mixture model agrees with the measured kappa. During the course of homogenization and formation of the gel state, kappa and C-p do not change significantly. The time for gel formation, t(gel), decreases rapidly when the sol is aged at high temperatures. The change occurs almost according to the relation, log(10)(t(gel)) proportional to 1/T. The accelerated formation of the Laponite gel at high T is distinguished from that of organic, mostly protein gels which form more rapidly at low T. The gels are not thermo-reversible. We consider the consequences of our findings for the current understanding of the phonon propagation and electrostatic interactions between H2O molecules and Laponite disks.