I'm using AJAX mode and would like to allow BACK browser button in magento

Not needed since Improved navigation 2.5.8. version.

To solve this issue in Improved Layered Navigation extension, please open the file js\amasty\amshopby\amshopby-ajax.js

and change line

var amshopby_use_hash = false;

with

var amshopby_use_hash = true;