Unmanned aerial vehicle (UAV) networks are convinced as a significant part of 5G and emerging 6G wireless networks. UAV slicing is a promising proposal of converging different services onto a common UAV network without deploying individual network solution for each type of service. This paper is concerned with UAV slicing for providing energy-efficient and fair service coverage for enhanced mobile broad-band (eMBB) users (UEs). Aiming at physically configuring UAV slices, the UAV slicing problem is formulated as a time-dependent mixed-integer-non-convex programming problem with a goal of maximizing all UEs' data rates while minimizing UAVs' total transmit power. To mitigate this challenging problem, we first decompose the original problem into two time-dependent subproblems using a Lyapunov approach. We then derive the procedure of tackling the non-convexity and the mixed-integer property of the subproblems by exploring a successive convex approximate (SCA) method and an alternative optimization scheme, respectively. Based on the derived results, we develop an algorithm with provable performance guarantees to mitigate the two subproblems repeatedly.