Go through the doors and into the temple. There is a Club, as well as Green and Blue Chests should you need them, at the bottom of the stairs. Go through the next door and you'll be thrown into a fight with a Cerberus. The Cerberus likes to "burrow" into the ground, pop up and pound the ground, sending out a little shockwave; it tends to do this move a lot. It can also swipe at you with its one claw, as well as send out a projectile straight from its middle head.
Not long into the battle, mini-Cerberuses will spawn and annoy you. Just grapple these until you kill them, but concentrate on the Cerberus. Keep hitting and running until you see the red halo, then mash the button that appears to kill it. This will lower the barrier and allow you to progress.
There's two Red Orb Chests, one on either side of the central ramp you will be pulling back soon. Grapple the point and move back to make a ramp appear. Go up and to the left to find a strange device that when you step into it, everything around you will turn to stone. Go to the opposite end of the room and break through the small door here to find a Phoenix Feather and Gorgon Eye.
Go to the central door and attack it to break it open, grabbing either of the chests here if needed. Head down the only path you can go and through the door at the top of the stairs. Once the cutscene is over, step into the mist and go up the right set of stairs, breaking through the door at the end. In this hallway, you'll have to fight two enemies, so kill them to open up the side rooms.
The room in the back has two Red Orb Chests and a Green one. Go back into the hallway and grab the statue, dragging it into the front room. There's a small indent you can place the statue on the right side, so do that and hit the mist to make a platform fall on top of the statue. Use this to get to the ledge above and open the two Red Orb Chests, as well as grab the Gorgon Eye and Phoenix Feather.
Go back to the main room, hit the mist and go up the left set of stairs now. Break through the door and you'll encounter a new enemy in this hallway. Whatever you do, don't try to grapple her, as she will shock you pretty hard. Dodge the attacks and keep hitting her until you get the red halo, which will automatically kill her when grappled.
Grab the statue and drag it into the front room, placing it onto the middle switch. Hit the mist to make everything stone and move the statue off the switch. Head to the north side of the room and when everything turns back to normal, roll under the gate into the room. Here, grab the two Red Orb Chests and another Gorgon Eye and Phoenix Feather. Pull the switch to return to the previous room.
Go to the room in the back and grab the Red Orb Chest, then hit the mist to make the floor crumble. Jump down and go into the hallway to enter the Delphi Catacombs.
