We address the problem of network pruning for extending the service life of satellites in LEO constellations. Satellites in LEO constellations can spend over 30 % of their time under the earth's umbra, time during which they are powered by batteries. While the batteries are recharged by solar energy, the depth of discharge they reach during eclipse significantly affects their lifetime--and by extension, the service life of the satellites themselves. For batteries of the type that power Iridium satellites, a 15 % increase to the depth of discharge can practically cut their service lives in half. In this paper, we present the design and evaluation of two forms of network pruning schemes that reduce the energy consumption of LEO satellite network. First, we propose a new lightweight traffic-agnostic metric for quantifiying the quality of a frugal topology, the Adequacy Index (ADI). After showing that the problem of minimizing the power consumption of a LEO network subject to a given ADI threshold is NP-hard, we propose heuristcs to solve it. Second, we propose traffic-aware metric for quantifiying the quality of a frugal topology, the maximum link utilization (MLU). Also, with the problem being NP-hard subject to a given MLU threshold, we propose heuristics to solve it. We evaluate both forms using realistic LEO topologies and traffic matrices. Results show that traffic-agnostic pruning and traffic-aware pruning can increase the satellite service life by as much as 85 and 80 %, respectively. This is accomplished by trading off very little in terms of average path length and congestion.