And now to head to the last temple (and the hardest): Damocles'.
In order to get to the entrance I suggest you use the same trick as before, just jump to the entrance. Lara won't die and it will save you of much more work.
Now, when you get to the gate, you'll notice it will start to fall as soon as you hang on the ledge. The idea there is to use the switch, then climb on the rock and when the gate reaches the top level (or somewhere up anyway) jump (the gate will start to fall) and immediately jump again. You need to be fast enough to still reach the ledge though. Basically, the moment you see Lara grabbed the gate jump again.
Alternative: (thanks to Domsy)
The idea is to raise the gate, and then push the block in between the spikes before it drops back down. That way the gate is stuck at the top, so you can easily jump up onto the top piece.
Now, this part of the level is all about swords. In all sizes and shapes (well, mostly HUGE). Admire the view, then just go on and pick up the key: The Key of Damocles!
That was pretty easy, right? Wrong. Move quickly before you are shread to pieces.
Explore the room a little big, there is some ammo and medikits around and then use the grapple to get to the top part.
Head to to switch (yes a bit of jumping involved) and flip it.
Now you can head to the other room.
You'll notice sword falling out of the ceiling. They are placed in the middle of the squares that form the ceiling. If you stay close to the wall or look up and avoid them, you should be safe. Eventually you can just look at the ceiling and dodge all of them once so you don't have to worry about them all the time.
Now you need to get across the room with the blades coming out of the floor. Fun, eh?
Just watch the pattern and move using the ALT key. At the last step, make sure you walk normally and jump.
Go to the broken column and jump to the ledge on the wall. Then to the upper ledge and finally to the switch on the left.
Next, jump towards the white column to the right. Lara will grab it and climb. Go up as you can, position with the back to the nearby column and do a back jump to it.
Jump to the other column and then to the small enclosure on the other side with the switch.
Do a running jump to the ledges to the right, then jump to the upper one, jump towards the ring and use the Q key to do a wall run. Then when you get as far as you can, jump towards the column nearby. This can be quite tricky since the key you need to press might be Back when you jump, it just need to be towards the column and not forward.
After all this (hopefully you didn't lose all your hope by now), jump from one column to another and then to the ledges on the wall and in the end to the room with the wheel.
Turn it as much as you can and hurry to through it since it closes automatically.
Phew, that was hard.
Anyway, head out and then down to the four locks. Use the keys in each of them and finally you can enter the Coliseum.