When using this token in a block, there's always the first replacement rendered, no matter which page i open. I believe, it's because the block isn't part of the current-page entity.
Add this token in a block, which is rendered on several pages. The first page loaded after a cache rebuild will serve the cached value for all other pages.
Add the cache context 'url.path' in hook_tokens of current_page_entity_tokens.module.