Are you trying to do database loadbalancing? Or just add another device inbetween app and db?
The pros and cons of database loadbalancing are different for different database management systems and also they are defined by the application.
Only lookig at SQL databases, they are just different between Microsoft SQL, MySQL and Oracle. Some of them come with loadbalancing solutions from the vendor, others have third party vendors offering loadbalancing solutions.
Loadbalancing must be supported at the application layer too. Read operations can be loadbalanced more easily, write operations might be more difficult. Think of consistency, how are db updates synchronized in case you send writes to different SQL servers?
But also, does the application allow read operations from database servers that have not yet been synchronized with the latest write update?
If you are looking at loadbalancing NoSQL solutions, other challenges might arise.
Take a look at this article. 9 years old, still relevant:
https://devcentral.f5.com/s/articles/the-challenges-of-sql-load-balancing