Fiber Fabry-Perot tunable filter (FFP-TF) is one of the core components of the demodulation system of a fiber Bragg grating (FBG) sensor. Its stability is essential to demodulation accuracy improvement, and temperature drift is one of the key factors affecting its stability. Given the problem of the transmission wavelength drift of FFP-TF in a variable temperature environment, a temperature drift compensation method based on the integrated moving window is proposed in this paper. The nonlinear relationship between transmission wavelength and temperature is modeled by the least squares support vector machine, and a moving window is built in the training samples. Moreover, a weighted integration method of the compensation result of each moving window through the neural network is proposed to mine more process information of the training samples. The experimental results show that the maximum measurement error of demodulation is ±13.5 pm when the integrated moving window is not introduced, and that is ±0.82 pm after introducing the integrated moving window, which means the proposed method effectively improves the temperature stability of the tunable filter demodulation in fiber grating sensing.