This paper investigates the optimal design of wireless powered fog computing networks, where a hybrid access point integrated with fog computing function (F-HAP) first charges an Internet of Things (IoT) device via wireless power transfer (WPT), and then the IoT device uses the harvested energy to compute real-time updates locally or offload the updates to the F-HAP for computing. For such a system, an age-energy aware cost function is defined to evaluate the system performance, based on which, an optimization problem is formulated to explore the minimum age-energy aware cost by jointly optimizing the time assignment, the transmit power, the computing frequency, as well as the computing mode selection, such that the given data processing task can be completed. Since the problem is non-convex with the discrete binary variable, variable substitution and Karush-Kuhn-Tucker (KKT) conditions are applied to solve it and some closed-form results on the optimal solution are derived. Numerical results show that the transmit power has much greater effects on the age-energy performance of the fog offloading mode than on that of the local computing mode. Moreover, when the IoT device is relatively close to the F-HAP, fog offloading is a better choice; Otherwise, local computing should be selected.