Since version 1.1.1 of Lesti::Fpc I get more support requests for the layered navigation. I did change the behavior of key generating in Lesti::Fpc to get more hits in the cache. I added the new field Uri Params in the configurations and wrote a post about it. This post is all you need to get your layered navigation work, but I guess it would be better to explain it again an example.
Who does magentos request work?
Zend Framework and Magento work with a key-value pair for parameters in the request. But at first magento uses rewrites to match for example
http://fpc.dev/catalog/category/view/id/3?price=1-2. With this uri magento builds the request. Here is a little diagram.
Why does layered navigation don’t work in default installation?
To produce more hits in the cache, we want to ignore senseless parameters. Only if you tell Lesti::Fpc that a parameter is important, it won’t be ignored. The attributes of your layered navigation aren’t in the default Uri Params. You have to add them to the configurations. In this example I have to add price.