Can landscape adhesive be used for a masonry fire pit?
Regarding Loctite PL 500 landscape adhesive (mentioned in the asker's answer)...

Service Temperature: -20°F (-29°C) to 150°F (66°C)

No, landscape adhesive (or at least that specific product) isn't rated for temperatures encountered in a fire pit.
That said, if you put a bead toward the outside of the course rings I'd expect it to hold up fairly well. Temperatures out there may get high enough to soften it, but probably wouldn't burn it.
Of course, there may be landscape adhesives (or products that could serve as such) with higher temperature tolerance.