Amasty Full Page Cache uses its own Table called core_cache. This database will bloat as visitors visit the website. It is a big flaw with the software. Our database increased from 300Mb up to 8.5GB which resulted in slow database transactions (i.e. add to cart, login, etc), and we have a website with approx 1,000 products.
The table core_cache stores all the timestamps for cached files (in /var/cache)
This will empty when you manually empty the cache. (System > Cache Management > Empty Cache)
But will not empty with a cronjob.
You need to empty it daily manually, or via a handwritten script to truncate the table.
Having Spoken to the Amasty Tech Support, you have to install Redis and configure it so that Redis does all the caching!
Or if you are on web hosting with no Redis (i.e. 1&1) you need to keep truncating the database. There is no other solution around it.