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Energy-efficient software helps improve mobile device experiences and reduce the carbon footprint of data centers. However, energy goals are often de-prioritized in order to meet other requirements. We take inspiration from recent work exploring the use of large language models (LLMs) for different software engineering activities. We propose a novel application of LLMs: as code optimizers for energy efficiency. We describe and evaluate a prototype, finding that over 6 small programs our system can improve energy efficiency in 3 of them, up to 2x better than compiler optimizations alone. From our experience, we identify some of the challenges of energy-efficient LLM code optimization and propose a research agenda.