Numerous Internet of Things (IoT) applications are sensitive to the freshness of status information for accurate monitoring and control. Due to the limitation of computational resources, status updates of IoT devices can be offloaded to the mobile edge computing (MEC) server for execution to enhance the information freshness. In this work, we investigate the computation offloading problem in the MEC-assisted IoT. Age of processing (AoP), which takes the data processing time into account, is adopted as a performance metric to measure the freshness of the information of IoT devices. To achieve the age-optimal solution, we explicitly consider the heterogeneity of the status information and develop the offloading decision with both linear and non-linear age functions. An average AoP minimization problem is then formulated as a decentralized computation offloading game and we propose an age-optimal computation offloading mechanism to obtain the optimal offloading strategy. Simulation results show the convergence of our proposed method and demonstrate the effectiveness of the proposed scheme under different scenarios.