Well the short answer is, yes. The temperature in your room, if using a air cooler, is the air your PC uses to cool the system. Cooler room, cooler system, hotter room... etc.
I would suggest at least popping the side off and taking a look that all the fans are working properly and clean out any dust within the sytem to give it optimal air flow with this increasing heat.
If your system is pretty clean then I would suggest getting a after market cooler. I just picked on up for 16.99 called the Cooler Master Hyper T2. This is better than stock cooler and fits most smaller cases.
It depends on what you do on the PC if this would be a worthwhile upgrade. If your just doing some light/medium gaming or some light overclocking this cooler...