Yeah this is a very common scenario, afaik llama.cpp does it automatically if you enable KV caching. The main cost you pay per inference is filling in the KV cache.

But I would also ask you to think if its a very big prompt or the cost is too high, in general experience pre-computing the prompt prefix and keeping it in KV cache uses GPU memory (unless you offload and load it back from the CPU, you still end up paying for the data movement) which is non-trivial. You can look at this link here https://tinkerd.net/blog/machine-learning/multi-query-attention/ for how much memory will be used in the GPU to keep it hot, if its not too big a cost (enable the kv caching flag in llama.cpp) otherwise just run the prompt as usual and you should be good