Low earth orbit (LEO) satellite network has been more and more widely used because of its advantages of low delay, low overhead, and flexible networking. However, the reliability and effectiveness of data transmission are affected by the specific characteristics of LEO satellite networks (LSN), such as time‐varying topology, uneven distribution of service demands, and multiple types of service. In this article, a service‐oriented routing with Markov space‐time graph (SOR‐MSG) in LSN is proposed to improve the reliability and effectiveness of information transmission. First, for the time‐varying topology, a Markov space‐time graph (MSG) model is constructed by calculating the connectivity time and connectivity probability between satellite node states. Then, in light of the uneven distribution of service demands, the space‐time factor (STF) is defined to balance the network load by scheduling the intersatellite link with lower loads. After that, we formulate the general routing problem caused by different quality of service (QoS) requirements for multimedia services and uneven load distribution and propose a multi‐QoS optimization objective function according to delay, remaining bandwidth, packet loss rate, and STF. Following this, based on the established MSG, the state transition rule and pheromone updating rule are optimally adjusted to select the next hop node, so as to finally determine the best information transmission path. Finally, the simulation results show that the SOR‐MSG algorithm can not only adapt to dynamically changing topologies and meet multiservice QoS requirements but also well balance the network load.