Real-world communication networks are complex networks, but designing load-balancing routing strategies in complex networks is not an easy task, especially when the traffic demand is under uncertainty. To hedge against such uncertainty, we propose a demand-oblivious routing strategy in complex networks (DOR-CN). Unlike traditional optimization-model-based methods, DOR-CN decouples the routing calculation with traffic demand by a randomized routing algorithm following the oblivious routing paradigm, and is able to achieve load-balancing without knowing the accurate traffic demand. In order to avoid the embedding distortion of complex networks, DOR-CN transfers the metric space of classical oblivious routing into the hyperbolic space. Theoretical derivation can prove that for arbitrary traffic demand matrix, the network congestion of DOR-CN routing strategy is under guaranteed. Experimental results demonstrate that DOR-CN can not only reduce network congestion, but also achieve better load-balancing performance and lower risk of route failures.