Can I mount an 80" TV where only one stud is available?
Above your fireplace the stud likely doesn't continue to the floor. Most likely it's atop a header above the fireplace. Now, I'd guess the "header" is just another 2x4, flat, spanning the space of the fireplace... not a "real" header.

This means that you don't have much downward support for the weight.

In my opinion, if the TV is not going to be on a moving, tilting or swiveling mount it will probably be ok with the following process:

Precisely locate the stud using small finishing nails tapped in to locate the edges. This is to get you into the true center of the stud. Take a sheet of hard-wood ply (I like birch) the full concealable height and width of the TV. This will accept multiple pre-drilled lag bolts directly into the middle of the stud. I'm nervous so I'd use three of them. 

Then I would glue the plywood into place against the drywall and bolt it into the stud. Let it cure, then attach the TV mount to the ply.

Primarily, though, I would wonder why there's only a single stud above the fireplace!