Buffering Elasticsearch results in (non)sql database

We have some requests to elasticsearch that appear much more often than others.
For example when a user enters our recipe section, we always show some predefined categories, which when clicked generate a Elasticsearch query. On the other hand users can search for random terms.

Is there a benefit to pre calculate the standard queries and serve them from another database?
My thinking is, that hosting Elasticsearch is quite expensive compared to another database on AWS.

Which one?

I thought in general, since I need more RAM/CPU for the same number for request, but I might be very wrong.

I would suggest you have a proper cache in place, for example Varnish to solve your issue. To me your approach is not a question of using any other datastore really. Elasticsearch is basically fast enough in most cases.