In order to explore the partial mechanical properties of functionally graded beams under thermal conditions, the axial deformation field of functionally graded beams was established by using Gauss-Lobatoo node and Chebyshev polynomial. The discrete governing equation of the axial deformation field under thermal conditions was derived by Chebyshev spectrum method and Lagrange equation. By applying boundary conditions to the model with projection matrix method and using the comparison results of uniform Timoshenko materials for auxiliary verification, the changes of the natural frequency of the system under different temperature fields, material gradient index and anchorage conditions were analyzed. At the same time, the thermal buckling temperature of the structure was studied. The results show that under the same conditions, the natural frequencies of each order of the system decrease with the increase of temperature, and the first-order frequency decreases the most. The fourth order natural frequency decreases the least. The variation trend of natural frequency of cantilever beam is the same as that of material softening, and the decrease of the third-order frequency is more obvious than that of the first-order frequency. The natural frequencies of each order of the structure can be optimized by adjusting the material gradient index. With the increase of material gradient index, the thermal buckling temperature of the material increases continuously, but for different functionally graded materials, the thermal buckling temperature changes show different trends. This study can provide support for the research and engineering application of functionally graded materials.