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LEO satellite networks possess highly dynamic topologies, with satellites moving at 27,000 km/hour to maintain their orbit. As satellites move, the characteristics of the satellite network routes change, triggering rerouting events. Frequent rerouting can cause poor performance for path-adaptive algorithms (e.g., congestion control). In this paper, we provide a thorough characterization of route variability in LEO satellite networks, focusing on route churn and RTT variability. We show that high route churn is common, with most paths used for less than half of their lifetime. With some paths used for just a few seconds. This churn is also unnecessary with rerouting leading to marginal gains in most cases (e.g., less than a 15% reduction in RTT). Moreover, we show that the high route churn is harmful to network utilization and congestion control performance. By examining RTT variability, we find that the smallest achievable RTT between two ground stations can increase by 2.5x as satellites move in their orbits. We show that the magnitude of RTT variability depends on the location of the communicating ground stations, exhibiting a spatial structure. Finally, we show that adding more satellites, and providing more routes between stations, does not necessarily reduce route variability. Rather, constellation configuration (i.e., the number of orbits and their inclination) plays a more significant role. We hope that the findings of this study will help with designing more robust routing algorithms for LEO satellite networks.