Download Update's fail (Unix File Permissions on /tmp directory)
You probably have your own tmp directory, which is not /tmp but /some/path/to/your/directory/tmp. You wouldn't be allowed to create /tmp2 for sure :)

To find that out, go to admin/reports/status/php and look for _SERVER["DOCUMENT_ROOT"], that will tell you where Drupal is located and from there, you should be able to build the path to your tmp directory.

Then set the temporary directory to that.