Today, I was playing a game of RollerCoaster Tycoon 3 on my new HP Laptop (Specs Below), when I realized the temperature of my CPU was approximately 82 degrees Celsius. Normally, my CPU (When I'm just surfing the web) stays around 50-60 degrees. This is an AMD Athlon II P340 Dual Core Processor clocked at 2.2ghz. I feel that this is a little odd, since it is a fairly big jump, and I feel that is a bad temperature to be running at, considering this game doesn't require much to run (It runs great on my old Celeron computer with 512mb of ram). Are these temperatures safe, and will it fry my hardware if I'm playing at these temperatures?

Specs:
HP G62-347NR Laptop
3GB RAM
AMD Athlon II P340 Dual Core Processor
Windows 7 Home Premium 64bit
AMD M880G with ATI Mobility Radeon HD 4250
320GB Seagate SATA Hard Disk

No that temp probably isn't safe. I'm surprised that it doesn't shut off. Mine does at 75 C. You need to open it up and clean it out with compressed air.

I just took my temperatures again (normal temps) and it stays around 60. This is an AMD Athlon X2 Mobile, and somewhere I saw the highest it can resist is up to 95 degress celsius. When I was playing, the max it went up to was 82 degrees, which seems ok for a high end game, but not a low-medium-end game like RCT3. I would clean it out, but 1. It is a laptop, and I'm not good at opening the case, and 2. This laptop has only been used for a little over a month. It should not need to be cleaned out already....

and will it fry my hardware

Hard drives are pretty robust. I would be more concerned with the pure electronics devices (CPU, motherboard, graphics, RAM, etc.).

Being one month old, you are right, it should not be full of heat trapping dust - yet. And while notebooks are not designed so normal users can open the case for proper cleaning, you can remove all cavity covers, open all bays, and remove the battery to gain some access to blast with a can of compressed dusting gas.

Note that in spite of what notebook marketing departments would have us believe, NO notebook makes a good gaming machine. Notebooks by their very nature are lightweight compact portable devices designed for the road-warrior needing to do email and conduct Power Point presentations while traveling. Their compact design does not allow for adequate cooling (a challenge even for full tower PCs) and as noted here, they do not allow for proper user cleaning. Assuming your fans are working properly, you may just be running into the limits of the notebook.

You might consider a Notebook Cooling Pad w/ext. power supply.

You are right about how notebooks are not meant for gaming, digerati (cool username, by the way). What I am kind of concerned about, is even performing simple tasks like typing this post right now, it sometimes jumps of to 70 degrees, and other times, it'll stay at around 50. It is pretty random. Also, when the CPU does go up to around 80, the game doesn't slow down or anything, it just keeps doing what its supposed to with no lag what-so-ever. I've been trying to increase the internal fan speed with Speedfan, but it keeps crashing (it's not related to the heat, however). I do have a cooling fan that is USB powered (but I plug it into the wall with my USB Adapter).

I also did some research on my CPU and I found one resource that said lower than 95 degrees is fine for my CPU. Could it be possible this just is my normal temperature?

Jumping up as long as it soon comes down is probably not a problem. If it stays up there longer than a minute or so, I would be concerned. Note that while the CPU may not suffer, that much heat building up inside the case may cause stability issues with other devices. And extended periods of high heat can stress the socket as well. Speedfan is decent program but I've never used it to control fan speed on a notebook. I generally rely on the maker to know what's best since they have [supposedly] chosen all the parts to work well together.

Note that when you are doing simple tasks, Windows may see that as idle time and choose to use that time for system housekeeping or other chores.

I think it is wise to plug your extra cooling fan into the wall. I find it odd to use a cooling device that draws power (thus creating more heat) from the very device you are trying to cool down.

cool username, by the way

It's a real word, BTW. An old commander nicknamed me that long ago when I saved his rear by fixing the General's computer. It just stuck after that.

I don't think I'm going to worry about it too much. Also, I should mention that I am running in High Performance mode, so maybe that is a factor. My friend had a similar laptop that I have and it always ran hot, so I'll ignore it, but if problems arise my HP warranty will cover it, right?

Well, don't ignore it - but don't let it drive you crazy either.