A few bits of clarification:
These files aren't in the WordPress tmp directory, they're in the w3tc tmp directory (wp-content/cache/tmp).
The files appear to be 0-bytes in length.
The file names are prefixed with "temp" and then a random string of 6 characters.
I checked the Apache error logs and there's nothing unusual/related there.
I wrote a quick PHP script to see if the web server was capable of chmod'ing and unlink'ing these files, and it seems to be able to.
One thing that's odd about the host is that they require the script that runs to have the same group as the file that it is writing. I don't know what PHP scripts are creating these files (admin? page loads?) but if they're being created by one and unlinked by another, that could be a problem in this environment.