This might be a good question for a granite shop. Many of those people know the qualities of other types of stone. My granite guy was taught by his father who was an old world stone worker. He can tell you all the qualities of different stones.

I know that granite is good to use for this purpose.

Travertine is a type of limestone, IIRC.

It's relatively porous, and isn't nearly as hard or stable as marble.

It may not be the best choice for what would be a high/extended heat application.

Travertine tyrpically has voids which are filled with an epoxy material at the factory. Exposing it to the expansion and contractions caused by the temperature swings in a fireplace will likely cause the expoxy to come out over time (due to the different rates of expansion between the expoxy and the stone) leaving voids. That said I have had travertine facing on my surround for some time and it is fine, but only the edge is near the opening and it doesn't wrap around into the firebox itself. Even a solid igneous (volcanic) rock like granite can crack when exposed to the direct heat of the fire (especially if there are veins present) which is why fireboxes are lined with specially made fire brick. Why do you need to wrap the facing into the firebox by 3" to 4"?

I had a fire pit built, the top lined with 2 inch travertine treads.. The entire top (octagon shape) cracked. I need to replace them with another stone.

The stone cannot withstand the heat