I think the problem lies in line 103 of warm-cache.php. It uses the command wp_generate_password() to create the variable plugin_warm_cache_api, which is the warm_cache=URL . wp_generate_password() by default uses special characters to be more secure, but these characters don't seem to work with cron or in a URL format in general.
In my case, % and ) caused either cron to fail, or the URL to just resolve my homepage. You've got a whole cl&sterf*ck.
Anyway, I changed my plugin_warm_cache_api variable manually in the database, and added $special_chars = false to the wp_generate_password() command for good measure. Seems to have done the trick.
add_option('plugin_warm_cache_api',wp_generate_password($special_chars = false));