Poly-ether-ether-ketone (PEEK) can replace traditional materials such as metals and ceramics in many fields and is widely used due to its excellent properties such as heat resistance, corrosion resistance, radiation resistance, fatigue resistance, and electrical insulation. Especially with the development and application of 5G technology, PEEK has become a popular material for 5G. Temperature is an important and key factor to affect the application of PEEK materials. This work studied the Terahertz (THz) spectroscopic characteristics of PEEK and their dependences on the temperature. It is using terahertz transmission spectroscopy, combined with a temperature control device, THz time-domain spectral signals of the PEEK flake sample were measured every 5 ℃ in the temperature range from 25 to 300 ℃ with a constant temperature increasing speed. THz absorption coefficient, dielectric constant and other optical constants of the PEEK flake can be obtained with the optical constant extraction algorithm. The temperature dependence of these THz spectroscopic parameters on the temperature was analyzed. In the effective spectral range of 0.5~4 THz, the experimental results show that at room temperature (25 ℃), PEEK has a distinct characteristic absorption peak at 3.5 THz. At the temperature range of 25~300 ℃, at 1 THz frequency, the absorption coefficient and the dielectric constant of PEEK have a fluctuation of 4.38% and 5.0%, respectively, relatively to room temperature. At room temperature, the PEEK at 1 THz has a dielectric loss tangent value of 2.5×10^-3. Compared to PMMA, PE and other polymers, the dielectric loss tangent value of PEEK is much lower; At the temperature range of 25~300 ℃, it remains relatively stable with a small fluctuation during heating, indicating excellent thermal stability and low dielectric loss of PEEK. The results in this work show that terahertz spectroscopy can be combined with a temperature-controlled device to study and characterize the thermal stability of polymer materials through the optical constants of the materials and obtain the dielectric properties of the materials at different temperatures. Terahertz spectroscopy is fast, efficient, label-free and non-destructive, and it can be used to study the internal defects, stability, and identification of materials. Simultaneously, the test data in this work can provide a reference for PEEK material applied in 5G, 6G, and other high-frequency communications at different temperatures.