My best guess is that you have a permissions issue when uploading the file. The installer puts it in that temp directory, and then tries to unzip it. If you don't have the permissions for Apache setup correctly on your site, then it may not be written properly to that /tmp directory.
To fix temporarily, I'd recommend unzipping it locally and FTP'ing it to the /tmp directory in your Joomla install. Then, you can use the "Install Directory" option instead of upload and install.
To see if my guess is correct, try going to "Help->System Info" and then clicking the "Directory Permissions" tab. Everything in there should be green and 'writeable'. If not, you'll need to fix those issues on your server.
Let us know if this helps or at least if it helps narrow down the issue.
Error during install
