Unmanned aerial vehicles (UAVs) are employed as mobile relay nodes to enable timely remote monitoring by collecting information from monitoring devices and transferring the collected information to base station. The freshness of delivered information is critical to system performance, which can be quantified by the concept of age of information (AoI). Nevertheless, the fresher information comes at the cost of higher energy consumption at UAVs. Considering the limited onboard energy, it is essential to strike a balance between the age of delivered information and the required energy budget. here, both straight trajectory and circular trajectory of UAV are considered, and study a problem with the goal of supporting timely data collection while minimizing the energy consumption at UAV. The problem is formulated as a multi‐criteria optimization problem by jointly considering the aging of collected information, the trajectory planning of UAV and energy consumption at UAV. To solve the formulated problem, a solution procedure to find a sequence of Pareto‐optimal points is proposed. Simulation results demonstrate the Pareto‐optimal curve, which yields the energy‐efficient UAV trajectory for timely data collection.