Surface temperature distribution is crucial for thermal property-based studies about irregular asteroids in our Solar System. While direct numerical simulations could model surface temperatures with high fidelity, they often take a significant amount of computational time, especially for problems where temperature distributions are required to be repeatedly calculated. To this end, deep operator neural network (DeepONet) provides a powerful tool due to its high computational efficiency and generalization ability. In this work, we applied DeepONet to the modelling of asteroid surface temperatures. Results show that the trained network is able to predict temperature with an accuracy of ~1% on average, while the computational cost is five orders of magnitude lower, hence enabling thermal property analysis in a multidimensional parameter space. As a preliminary application, we analyzed the orbital evolution of asteroids through direct N-body simulations embedded with instantaneous Yarkovsky effect inferred by DeepONet-based thermophysical modelling.Taking asteroids (3200) Phaethon and (89433) 2001 WM41 as examples, we show the efficacy and efficiency of our AI-based approach.