Recommendation systems that use auxiliary information such as product names and categories have been proposed to address the cold-start problem. However, these methods do not perform well when we only have insufficient warm-start training data. On the other hand, large language models (LLMs) can perform as effective cold-start recommendation systems even with limited warm-start data. However, they require numerous API calls for inferences, which leads to high operational costs in terms of time and money. This is a significant concern in industrial applications. In this paper, we introduce a new method, RevAug, which leverages LLMs as a data augmentation to enhance cost-efficient cold-start recommendation systems. To generate pseudo-samples, we have reversed the commonly used prompt for an LLM from ``Would this user like this item?'' to ``What kind of items would this user like?''. Generated outputs by this reverse prompt are pseudo-auxiliary information utilized to enhance recommendation systems in the training phase. In numerical experiments with four real-world datasets, RevAug demonstrated superior performance in cold-start settings with limited warm-start data compared to existing methods. Moreover, RevAug significantly reduced API fees and processing time compared to an LLM-based recommendation method.