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Routing is a challenging issue when dealing with the multi-hop cognitive radio networks (CRNs). The stability of a route is highly influenced by the behaviour of the primary user (PU). The more frequently PU changes its activity, the less stable spectrum availability the cognitive user has. This causes tremendous stability fluctuation of the route in CRNs. The radio environmental issues that affect the route stability for the CRNs are: primary exposed node (PEN) and primary hidden node (PHN). These issues degrade the route stability and are the most important problems in the CRNs. To address this stability problem, the authors propose a radio environment aware stable routing (RASR) scheme for multi-hop CRNs. In order to mitigate the PEN and PHN problems, a channel priority list is kept locally by each secondary user. RASR can evaluate the route stability and can find appropriate route by defining a routing metric, which uses the second smallest eigenvalue of the Laplacian matrix. The proposed routing metric takes into account the route stability, path length and the activity of the PU. Simulation result shows that the proposed RASR enhances the route stability compared with the conventional routing schemes ad hoc on-demand distance vector with hop-count routing metric, and connectivity-based routing scheme Gymkhana.