There are 2 solutions to this. The first and probably best is to go and purchase the Flamebreaker boots from the store at the Goron city. It's a bit of a pain, but you need 2 pieces of the armor to withstand the heat in this area. Since the helmet costs 2000 Rupees it's best to use the armor and the boots. Having 2 pieces equipped will provide enough of a heat resistance to withstand the heat at the abandoned mine.
The second solution is to use an elixir. The heat resistance elixir provides enough fire defense for you to be able to roam around the abandoned mine without any issues.
Since elixirs run out and the armor does not, id advise spending the money on the armor. It will last much longer. You may end up spending the same if not more on elixirs in the long run. It's cheaper to bite the bullet now and not have to deal with the pressure of time limits when you are in an area that is too hot for Link to handle.