Premature end of script headers: wp-cron.php in WordPress 3.6

Please, check if this topic is helpful.

The Premature end of script headers: error is a very vague error that could be caused by numerous different things. It is an apache error and it could be related to a memory limit being hit. Some more info here, http://www.webnethosting.net/kb/common-apache-errors-premature-end-of-script-headers/

It might be a good idea to check with your host and see if they have any ideas. Since it is happening on wp-cron.php it could be caused by a scheduled task or wp-cron function that is scheduled to run at certain intervals. These processes run in the background so you won’t directly notice any failures except in your logs or the scheduled task not being done.