Alright, I have added an asterisk@* alias to the Addresses in Emails, which solved this problem.
It was caused by the server tried to send the following cron error message by email:
PHP Parse error: syntax error, unexpected 'class' (T_CLASS), expecting identifier (T_STRING) or variable (T_VARIABLE) or '{' or '$' in /var/www/html/freepbx/admin/libraries/Composer/vendor/symfony/translation/Translator.php on line 90
I tried to solve it with php scl, but no luck yet.
After adding scl enable rh-php56 to the crontab, it works one and then the crontab gets overwritten to the original value.
It looks like /home/asterisk/.bash_profile has no effect on cron jobs of asterisk. I tried the following there:
source scl_source enable rh-php56
Edit: reinstalled FreePBX and it looks like everything works now.