Vinegar has a good reputation as a natural cleaner, and on most surfaces it deserves it. On natural stone it is one of the most damaging things you can reach for, and the damage is not reversible by cleaning.
Travertine, marble, limestone, and onyx are all calcium carbonate. Acid dissolves calcium carbonate. When vinegar, lemon juice, or an acidic bathroom cleaner sits on one of these stones, it chemically eats a microscopic layer of the surface. The result is called etching: a dull, slightly rough patch where the polish used to be. It isn't a stain sitting on top of the stone, it's a hole in the stone's finish, which is why scrubbing does nothing and why no cleaner can wash it away. Fixing etching means re-honing and re-polishing the surface.
Plenty of ordinary products are acidic without saying so on the front of the bottle. Most tub and tile cleaners, anything marketed for soap scum or hard-water spots, descalers, and a lot of general-purpose sprays fall into this category. So do spills you might not think about — orange juice, wine, coffee, and tomato sauce all etch a polished travertine counter if they sit long enough.
The safe answer is a pH-neutral cleaner made for stone, used with a soft mop or cloth, and spills wiped up promptly rather than left to sit. That's genuinely the whole maintenance routine. Granite and most porcelain tile are far more forgiving, which is part of why the mistake is so common: people apply what worked on their old kitchen floor to a new travertine one.
Travertine has one more quirk worth knowing about. It forms with natural holes running through it, and most floor tiles are sold with those holes filled. Over time the filler pops out and the open holes collect soil, so a travertine floor can look dirty even after a thorough surface clean. Those holes need flushing out, not skimming over, and filler can be replaced afterward where it's missing.
If a stone floor has already gone dull or patchy, it's worth getting an honest assessment before spending money on more cleaning. Sometimes what's needed is cleaning and sealing. Sometimes the finish is etched and the answer is restoration. Those are different jobs at very different prices, and you should know which one you're buying.