This study presents a direct numerical simulation (DNS) of a transitional hypersonic boundary layer in chemical non-equilibrium, with a focus on the influence of wall conditions. The simulation investigates a Mach 10 flow over a flat plate, where the wall is modeled as cold and isothermal with two different wall temperatures. The high stagnation enthalpy leads to significant aerodynamic heating, activating chemical reactions within the boundary layer and making chemical non-equilibrium effects prominent. Laminar-turbulent transition is induced via a tripping volume force applied along a narrow strip of the wall. The objective of this study is to investigate how wall cooling influences the development and transition of the boundary layer, the chemical activity, and the stream-wise evolution of wall shear stress and wall heat flux.