In a recent paper by Zhang etal. in 2012, a Mach number-invariant scaling was proposed to account for the effect of variation of free-stream Mach number in supersonic turbulent boundary layers. The present work focuses on the effect of variation of wall temperature with strong heating and cooling at the wall. Direct numerical simulation is used to study scaling and turbulence structure of a spatially evolving Mach 2 supersonic boundary layer at a friction Reynolds number of 500. A new scaling law is proposed to account for temperature-dependent fluid-property variations. This universal scaling appears superior to the existing models with the novelty that it applies not only for the mean-velocity profile but also extends to the turbulent transport, production, and dissipation terms in the budget of the turbulent kinetic energy.