Thanks that’s what was tripping me up!
For anyone else I was able to get it working with a few minor adjustments:
<?php $julytime = mktime(0, 0, 0, 7, 31, 2008);
if (get_the_time('U') < $julytime) echo "July 2008";
else the_time('F d, Y', '', '');
?>
Took out the extra bracket in the mktime statement and modified the the date slightly.