Large language models (LLMs) have achieved remarkable success in natural language processing, yet their performance and computational costs vary significantly. LLM routers play a crucial role in dynamically balancing these trade-offs. While previous studies have primarily focused on routing efficiency, security vulnerabilities throughout the entire LLM router life cycle, from training to inference, remain largely unexplored. In this paper, we present a comprehensive investigation into the life-cycle routing vulnerabilities of LLM routers. We evaluate both white-box and black-box adversarial robustness, as well as backdoor robustness, across several representative routing models under extensive experimental settings. Our experiments uncover several key findings: 1) Mainstream DNN-based routers tend to exhibit the weakest adversarial and backdoor robustness, largely due to their strong feature extraction capabilities that amplify vulnerabilities during both training and inference; 2) Training-free routers demonstrate the strongest robustness across different attack types, benefiting from the absence of learnable parameters that can be manipulated. These findings highlight critical security risks spanning the entire life cycle of LLM routers and provide insights for developing more robust models.