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This paper focuses on energy efficiency (EE) pre-coding design and computational-efficient precoding updating strategy for low earth orbit (LEO) satellite communications. Firstly, we formulate the EE precoding problem, which aims to maximize the EE metric under the quality of service (QoS) constraint and per-antenna power constraint (PAPC). By intro-ducing semidefinite relaxation, first-order Taylor approximation, and quadratic transformation, the problem is transferred into a convex one that can be efficiently solved. Moreover, due to the continuous movement of LEO satellites, precoding is performed frequently to maintain the high EE performance, leading to high computational complexity. Consequently, we consider prolonging precoding intervals to reduce complexity while alleviating severe performance degradation during the intervals. To this end, a computational-efficient beam direction change (BDC …