A novel approach to perform linear stability analysis using a spectral element-based reduced order model is proposed, thereby facilitating the comprehensive study of (thermo)acoustic (in)stability across complex waveguide configurations. Providing a generalization of a wave-based method (e.g. spectral element method) to the Laplace domain, this study is offering not only a new methodology for analyzing thermoacoustic systems but also expanding the application of the spectral element method and other wave-based techniques to a broader class of problems. By solving the wave equation in the Laplace domain, spectral elementary matrices can be defined in the complex plane in both wavenumber and frequency domains, allowing for an examination of system stability. This technique supports a wide range of waveguide investigations, whether using an analytical description to get the spectral matrix or a numerical method to determine the dispersion curves and eigenvectors in the cross-section. Additionally, the proposed method simplifies the implementation of parametric optimization procedures due to its low computational cost, thus offering significant advancements in the study of waveguide behavior of thermoacoustic systems.