Hypersonic boundary-layer stability has significant importance in vehicle design and successful operation. This paper investigates the stabilization effects of local wall cooling on the hypersonic boundary layers over a 5 deg halfangle blunt cone with a nose radius of 0.0254 mm. We employed a high-order-accurate flow solver to calculate the steady flow for a freestream Mach number of 6.0 and a unit Reynolds number of 25.59×10^6∕m. In simulations, we considered partial wall cooling, entire wall cooling, and adiabatic wall scenarios. Furthermore, we examined partial cooling parameters such as strip location, length, and temperature profiles. We calculated the growth rates, phase speed, and N-factor diagrams using a linear stability analysis. The results showed that complete wall cooling destabilizes the boundary layer. However, the cooling strip upstream of the synchronization point stabilized the boundary layer by damping the disturbances. The longer cooling strip further stabilized the boundary layer. The cooling strip placed downstream of the synchronization point destabilized the boundary layer.