Adaptive robot perception systems must balance detector accuracy, inference latency, and energy consumption, particularly when high ambient temperatures cause processors to thermally throttle. We propose Thermal-Headroom Routing (THR), a lightweight model-selection policy that forecasts processor temperature using a closed-form RC thermal model and excludes detector configurations predicted to cross a throttling threshold. We evaluate THR with four YOLO11 configurations across seven real-world perception traces, eight ambient temperatures, and 20 randomized seeds, comparing it with latency-aware routing, fixed-threshold thermostats, duty caps, and a matched-compute-budget control. At ambient temperatures of 45-55 degrees C and a tight inference deadline, THR eliminates throttling in simulation, reduces deadline misses by 3.9-7.5 percentage points, and reduces energy per on-time frame by approximately 12-13% relative to a latency-belief router. A separate experiment using deadline-paced, nonoverlapping frame execution yields miss-rate reductions of 6.2-9.5 percentage points. A tuned thermostat or duty cap can match THR at a fixed operating point; predictive routing is most useful when ambient conditions or calibrated thermal parameters differ from those used to tune static policies. Measurements on an Apple M5 support the single-node heating approximation but do not validate hardware throttling. The results characterize the conditions under which thermal forecasting improves the efficiency and timeliness of resource-constrained robot perception.