Is it okay to put a hot pan on a granite counter?

In general, the chances of significant heat damage to granite are very small, though depending on the pan, the granite, and any products that may be on the granite (not only sealant, but residue from granite cleaning or polishing products, etc.), you might sometimes get some staining/charring.

I would also take into account the amount of heat that will transfer from the pan/pot/tray -- a small pan which is nearly empty and not very hot is a very different thing from a giant cast iron dutch oven filled with chili or superheated cooking oil. I personally wouldn't risk injecting a huge amount of heat into a small area, because it will cause the material to expand and contract unevenly. When done repeatedly, it could even lead to cracking. Unlikely? Yes. Impossible? No, particularly if the granite is thin or lower quality (e.g., with some minor fissures or weak spots already in the stone, etc.).

Another issue is a safety one -- granite countertops that absorb heat from very hot pans can stay exceptionally hot for a quite a while. Since there's often no visual cue about the temperature (unlike a stovetop), it can easily cause burns if you forget that that part of the countertop may be hot.

Other answers have mentioned that most references on the internet say it's safe, but with just a quick search in the first 10 links or so that came up, I see a number of places that actually manufacture or install granite do NOT recommend placing hot things directly on granite.

See, for example:

• here: "Trivets or cloths are highly recommended."
• here: "Repeated exposure to high heat can eventually leave black marks and gashes on granite slabs."
• here: "The darker the material the more dense and therefore the more heat it will hold. The biggest issue with putting a hot pan on granite is that if you touched the spot it could be as hot as the pan itself and therefore cause you an injury. You should especially be cautious with black granite since it is the most dense and with repeat exposure may actually crack."
• here in a list of myths: " 'Granite is heat resistant, so it is perfectly okay to place hot pots on a granite countertop.' While it’s true that granite is quite resistant to the sorts of temperatures encountered in kitchens, excessive heat can damage or discolor some types of granite sealers. Rapid heating can also generate internal stresses that could cause a crack at a weak spot, such as a natural flaw or fissure in the stone. An easy way to eliminate potential problems is by using a trivet with feet."

Many sites agree with these cautions. There are other sites that say it should be okay, or it's safe up to X degrees, or it's okay but you shouldn't do it repeatedly, or whatever. Everybody seems to have their opinion. And you can find sources that claim the cracking is a myth. This video, for example, heats small blocks of granite in a toaster oven and with a blowtorch and observes no damage. But the heat damage (if it exists) is not going to be caused simply by heating, but by thermal shock created when a large slab is heated unevenly and thus expands and contracts unevenly (usually repeatedly over a period of years). If you look around a bit, the internet does have anecdotes of people who have had granite countertops develop cracks or discoloration.

Personally, I have granite, and I avoid doing it on a regular basis. However, in a pinch when I have a lot of hot pans or trays around the kitchen, and I need to set something down, I don't worry too much about doing it once in a while. I think the chances of damage are quite small. But with the cost of granite, why take any risk when you can just use a trivet or something (as you'd do on just about any other surface)? That's just my habit anywhere.