For certain hypersonic flight conditions, high temperature effects, also known as "real gas effects", become important. In this context, chemical equilibrium refers to a flow where the time scales associated with the chemical reactions are much shorter than the characteristic time scales associated with the fluid dynamics. In this paper, the hydrodynamic stability of the compressible boundary layer in chemical equilibrium was investigated using linearized Navier-Stokes simulations including a simplified chemical equilibrium model. Adequate knowledge of the possible hydrodynamic instabilities present for hypersonic chemical equilibrium flows, is essential for the design of aircraft and engines that fly at such speeds. A high-order-accurate Linearized Navier-Stokes code was developed to investigate the stability of hypersonic boundary layers in chemical equilibrium. Towards this end, the conservation equations for a three-dimensional viscous compressible laminar chemical equilibrium flow subjected to infinitesimal disturbances were derived. These equations were discretized in space using 6th and 4th order finite differences in the streamwise and wall-normal directions, respectively, and integrated in time using a 4th order Runge Kutta scheme. For model verification, test cases for a flat plate were compared with DNS and LST reference data, and presented very good agreement for growth rates and eigenfunctions. The new Linearized Navier-Stokes method enables quantification of the instabilities, offers a cost effective way to investigate the linear instability regime, and the influence of compressibility and high temperature effects.