I'm arriving Frankfurt from SFO on a United Polaris Business Class ticket but departing for Bologna on Lufthansa economy (United Mileage Upgrade Award booked back on December). I have a seven (yes seven) hour connection (not originally planned that way). If I was on a full 1st/Business class booking I would expect to get lounge access, but I don't think I get anything with the outbound economy leg.
A reply to a query on Reddit indicated I may have access to the Lufthansa "Welcome" lounge, but that is outside security near baggage claim.
There are some third-party operators that you can use to get lounge access. "Priority Pass" is a membership deal and offers lounge access and/or dining credits at various airports.
Another I just learned about is Lounge Buddy. It looks like you can purchase one-time passes to various lounges. For example, a one-time pass to the Lufthansa business lounge at FRA is available for $70. https://www.loungebuddy.com/FRA?source=homepage#self
Anybody with experience with either of these outfits?