Accurate and rapid demodulation plays a crucial role in fiber Bragg grating (FBG) sensing systems. The Fabry-Perot (F-P) filter is a dependable demodulation technique with excellent accuracy. However, the F-P filter suffers from demodulation error drift in a temperature-changing environment. The non-repeatable wavelength scanning affects the accuracy and stability of demodulation. To solve the problem, an accurate and rapid calibration approach is proposed for FBG demodulation. First, the temporal convolutional network (TCN) is utilized to extract the hidden information and long-term temporal relationships in the input features including temperature, a temperature changing rate, and shift of reference grating. Second, a state-of-the-art light gradient boosting machine (LightGBM) capable of forecasting demodulation error is adopted for rapid forecasting. To demonstrate the effectiveness of the proposed approach, the traditional TCN model and the composited-residual-block TCN (CTCN) model are both discussed, and the temperature-drift experiments are designed and conducted in two temperature-changing environments. The experimental results show that, compared to the widely applied long short-term memory (LSTM) model, the TCN-LightGBM model achieves higher prediction accuracy and reduces computation time by at least 59.68%. The proposed approach is an affordable and effective alternative to the existing hardware-based calibration techniques.