<p>Many optimization problems are deeply connected to eigenvalue problems. Therefore, studying eigenvalues is essential for effectively solving some optimization challenges. This paper mainly investigates the perturbation analysis of generalized T-eigenvalues for third-order tensors within the T-product framework, with the aim of providing a fundamental tool for better understanding some tensor optimization problems associate with it. Based on the classical perturbation analysis for matrix pairs, we establish perturbation theories for generalized T-eigenvalues of third-order tensors, including various decompositions, especially the generalized Schur decomposition for both complex and real cases, the Gershgorin-type theorem for regular tensor pairs, the Bauer-Fike theorem for diagonalizable tensor pairs, and the Weyl-type theorem for definite tensor pairs. Additionally, we develop the pseudospectra theory for generalized T-eigenvalues of third-order tensors, detailing fundamental properties of backward errors and pseudospectrum for regular tensor pairs. Algorithms for computing various decompositions, assessing backward error, and plotting pseudospectrum contours are presented. Two numerical examples are also provided to illustrate the computational results of pseudospectrum.</p>