Greater rifts have their own difficulty and dont care if you start your game in torment one or six. But you keep your settings for gold and experience.
Example:
I played torment three and four the last days until I discovered this. I had about 3.5k gold find bonus. Then I started a game on torment six and have 6.6k gold find bonus. Basicly I did not change the difficulty of the greater rifts but I maximized or maybe doubled my gold gain at the end of the rift. It might not be much but for those that dont have much gold nor the hoarder gem might want the extra gold since the rift is not very rewarding when it comes to gold.