I particularly like using OCCT to test my CPU, GPU or PSU on my computer.
No installation needed for the latest version.
Is the HD5870 GPU Fan running when it starts to overheat? Do you have good air flow in the computer case. Try removing one side panel and see if it runs cooler.
Possibly the Thermal Paste on the GPU Heat sink is bad and not conducting heat as efficient as when it was new. I would look at YouTube on how to replace the old Thermal Paste on your GPU card to make sure you are able to do it.
Also make sure the Heat Sink Fins are clean and not full of dust.