SpaceX plans ambitiously to launch approximately 12,000 satellites from 2019 to 2024, expected to be a complement or even competitor to ground networks. However, the mega-scale satellite network is topology-varying and the frequency of inter-satellite link (ISL) handovers increases rapidly as the topology expands, which will further arouse a massive number of route updates with considerable packet travel delay or even packet loss during the route convergence. The classic Dijkstra's algorithm is adopted for space route calculation, however, it always selects the default shortest path from multiple equal-cost shortest paths between two satellite nodes. To reduce the route change as much as possible during the periodical topology change, in this work, we refined the original Dijkstra and propose StableRoute to select the most appropriate route from the equal-cost candidates with the least route updates compared with the routing table last round. In this way, the end-to-end paths can be maintained as far as possible without time-to-time oscillation. Evaluation shows that it reduces 41% of the route updates in a 36 × 36 topology compared with Dijkstra, and the reduction rate will rise persistently with the growth of the satellite constellation.