Compositional spatiotemporal reasoning often requires a system to invoke multiple heterogeneous specialists, such as geometric, temporal, topological, and trajectory agents. A central question is how such a system should route among specialists when execution does not simply succeed or fail, but fails in qualitatively different ways. Existing tool-augmented and multi-agent LLM systems typically leave this routing decision implicit in language generation, making recovery ad hoc, difficult to interpret, and hard to optimize. This paper presents STAR (Spatio-Temporal Agent Router), a failure-aware routing framework that externalizes inter-agent control as a state-conditioned transition policy over the current agent, task type, and typed execution status. At the center of STARis an agent routing matrix that combines expert-specified nominal routes with recovery transitions learned from execution traces. Because the matrix conditions on distinct failure states, the router can respond differently to malformed outputs, missing dependencies, and tool--query mismatches, rather than collapsing them into a generic retry signal. Specialists execute through a tool-grounded extract--compute--deposit protocol and write intermediate results to a shared blackboard for downstream fusion. Results prove that retaining unsuccessful traces during training enlarges the support of the routing policy on error states, enabling recovery transitions that success-only training cannot represent. Across three spatiotemporal benchmarks and eight backbone LLMs, STAR improves over multiple baselines with the clearest gains on queries whose execution deviates from the nominal routing path. Router-specific ablations and recovery analyses further show that typed failure-aware routing, rather than specialist composition alone, is a key factor for these improvements.