No, not be default. But it will get written to the homefs file directly if you enable home persistence. Changes will also get written to a file (the rootfs file) if you enable static root persistence which should work well on an ssd. If you enable both then homefs wins. With static root persistence, all file system changes are stored on disk, otherwise they are stored in RAM.
You can also have the persistence files written to a different device if you use the pdev= plab= or puuid= cheats.
(general firefox info, not specific to a frugal install)
You can control “does” or “does not” by editing the firefox preferences about:config
To completely disable:
browser.cache.disk.enable = false
If you have other drives attached, instead of disabling, you might specify a custom location (not on SSD)
browser.cache.disk.parent_directory = /path/of/your/choice
SEPARATE FROM ABOVE, firefox frequently writes session and history to disk.
The value for prefkey browser.sessionstore.interval = is given in milliseconds.
IIRC, the default is 15000 (15 seconds); I’ve set mine to 120000 (2 minutes).
To avoid these write operations from hitting the SSD, you can (create and/or) use a custom profile for which you’ve specified a profile directory on a separate drive Profile Manager managing profiles
or you might achieve a workaround by creating a mountpoint, or using a symlink…