I had the same exact problem last year using jute twine from Lowes. It broke several times and the hops suffered and I had to jury-rig it towards the end of the season because the bines had grown to the top. The package says "heavy duty" and "use for gardening", but my South Florida summer sun cooked it.
This past fall I had some new growth that I wanted to support to see what happens during our growing season down here so I set up a new trellis with some nylon construction twine, but the bines didn't seem to stick very well to it. I still have it up, but I've been looking for alternatives lately and recently grabbed some sisal twine. Problem is after opening the package it wreaked of fuel-oil. I've since read about some types being treated with a rodent deterrent, and sisal also has a sometimes pungent odor, but if it's petroleum based I don't want that leaching into my soil or touching my hops.
That's my experience. I'm considering the Coir Yarn from Freshops. That or some small gauge (3/8") sisal rope, but it only comes in 50' lengths locally and I don't want to spend $20 plus extra hardware to accommodate the rope just to string up my hops.