Secure routing algorithm plays an important role in the performance of LEO satellite network survivability. However, due to the instability, openness and exposure of inter-satellite links, the internal routing of LEO satellite network is vulnerable to malicious attacks. Considering that trust management has a better performance in solving the internal attacks, this paper proposes a secure routing algorithm based on node trust for LEO satellite network, called SLT. The SLT is based on the distributed trust evaluation model, which is used to calculate the direct trust, indirect trust and aggregate trust value between satellite nodes through D-S evidence theory. Then, combining the low cost OPSPF routing protocol with the trust evaluation, the SLT algorithm is designed. Through timely detection and isolation of malicious nodes in the satellite network, STL algorithm can reduce the impact of malicious packet dropping caused by internal attacks. Finally, the simulations verifies that SLT algorithm has an average increase of 27% in packet delivery rate and an average decrease of 70% in packet loss rate, compared with OPSPF.