To mitigate thermal error effects on the linear axis positioning accuracy of gear grinding machines and enhance gear tooth surface quality and consistency, a thermal error prediction method based on the golden jackal optimization (GJO) algorithm and long short-term memory (LSTM) neural network is proposed. Using the H650GA grinding machine’s linear axis as the research object, PT100 temperature sensors and an XL-80 laser interferometer were employed to collect temperature and thermal error data under various working conditions. The fuzzy exponent, iteration count, and termination conditions of the Fuzzy C-Means clustering algorithm were optimized using the genetic algorithm, and key temperature-sensitive points were selected through Grey Relational Analysis. The GJO algorithm was employed to optimize the number of hidden layer nodes, initial learning rate, and L2 regularization coefficient of the LSTM network. A GJO-LSTM thermal error prediction model was constructed using training set data. Finally, the predictive performance of the GJO-LSTM model was validated with test set data. Experimental results demonstrated that compared to GA-LSTM, LSTM, and BP models, the GJO-LSTM model reduces RMSE by 30.3%, 42.8%, and 54.8%, and MAE by 36.2%, 41.1%, and 45.4%, respectively, confirming its superior prediction accuracy and robustness.