Data packet routing in aeronautical ad-hoc networks (AANETs) is challenging due to their high-dynamic topology. In this paper, we invoke deep learning (DL) to assist routing in AANETs. We set out from the single objective of minimizing the end-to-end (E2E) delay. Specifically, a deep neural network (DNN) is conceived for mapping the local geographic information observed by the forwarding node into the information required for determining the optimal next hop. The DNN is trained by exploiting the regular mobility pattern of commercial passenger airplanes from historical flight data. After training, the DNN is stored by each airplane for assisting their routing decisions during flight relying solely on local geographic information. Furthermore, we extend the DL-aided routing algorithm to a multi-objective scenario, where we aim for simultaneously minimizing the delay, maximizing the path capacity, and maximizing the path lifetime. Our simulation results based on real flight data show that the proposed DL-aided routing outperforms existing position-based routing protocols in terms of its E2E delay, path capacity as well as path lifetime, and it is capable of approaching the Pareto front that is obtained using global link information.