If the system doesn't contain a temp/tmp directory, this appears to be a hosting configuration issue, so you'd have to take that up with your provider.
Your solution to create your own temp directory may work, but your path wouldn't be correct. "/" is only the root as perceived by your FTP session because FTP sessions are typically chroot'd. Your home's root directory as perceived by the web server is going to be your home directory, which may be something like /home/my-domain. You can use the PHP getcwd() function to find out exactly what it is.