Code:
<form method="GET" action="/timesheet/enter_times.php" name="pp" id="pp">
<label for="pay_period">Select the pay period to enter hours for:</label><br>
<select id="pay_period" name="pay_period" onchange="document.getElementById('pp').submit();">
<option value="1368360000" >May 13, 2013 -> May 19, 2013</option>
<option value="1368964800" selected="selected">May 20, 2013 -> May 26, 2013</option>
</select>
</form>
Note: Added id to form and used document.getElementById to get, then submit it.