Large language models (LLMs) exhibit substantial variability in performance and computational cost across tasks and queries, motivating routing systems that select models to meet user-specific cost–performance trade-offs. However, existing routers generalize poorly in cold-start scenarios where in-domain training data is unavailable. We address this limitation with a multi-level task-profile–guided data synthesis framework that constructs a hierarchical task taxonomy and produces diverse question–answer pairs to approximate the test-time query distribution. Building on this, we introduce TRouter, a task-type–aware router approach that models query-conditioned cost and performance via latent task-type variables, with prior regularization derived from the synthesized task taxonomy. This design enhances TRouter’s routing utility under both cold-start and in-domain settings. Across multiple benchmarks, we show that our synthesis framework alleviates cold-start issues and that TRouter delivers effective LLM routing.