Low earth orbit (LEO) satellite networks can provide Internet service to users in areas where cellular networks are difficult to deploy. One critical function of satellites is to transfer data from satellite networks to ground core network through earth stations (ESs). The Ka-band multiple-input multiple-output (MIMO) can be used to establish feeder links with larger bandwidth between satellites and ESs. However, propagation at the Ka band is subjected to rain attenuation and various atmospheric fading mechanisms, which severely reduce the maximum link capacity. As a result, the insufficient capacity of feed link becomes the throughput bottleneck of satellite networks. In order to increase network throughput, it is important to fully use feeder link resources. In this paper, we propose a cooperation scheme to route packets to ESs through a well-resourced feeder link, such that the bandwidth of the feeder links can be fully utilized and the throughput of data downloading at the ESs is maximized. Firstly, we model the satellite network system and the feeder link based on MIMO technology. Then, a Maximum-Flow-Minimum-Cost (MCMF) routing algorithm consisting of two Linear Programs (LPs) is presented to compute maximum-flow routings for data download. Eventually, a variety of simulations are conducted to assess the proposed scheme, which shows that the cooperative routing scheme outperforms the existing SiRRS method in terms of throughput and delay.