These screenshots show the single-player game, but other than the boss being a little tougher and there being more minor enemies in the parts where they appear, the battle is essentially the same in all versions.
After topping up at the AMMO CACHE near the entrance, step forward and use the Staff of Osiris to rotate the 2 huge gears so you can jump past them to the other side of the room. Approach the alcove on the far left to trigger a brief cutscene.
The Pharaoh appears, screaming like an angry teakettle. Now the fight begins.
You may want to start by planting a remote bomb near the edge of the spike pit between the 2 gears. When the Pharaoh jumps across this pit toward you...
...you can immediately do a little damage by detonating the bomb as soon as he lands. Bombs don't hurt him much, though, so you'll need to concentrate on using weapon fire to chip away at his health.
Unlike Khepri and Sobek, there is no "trick" to beating this boss. It just takes agility and firepower. Watch the floor throughout the fight. It is completely covered in ice vents which flare up at intervals. Each vent glows white just before it activates, spewing a blast of frigid air. Try to move out of the way before that happens. Even with cold resistance, you'll still take damage if you stand on or jump over an active vent, but you will not be slowed by the cold.
Here, Lara runs over over the vents as they are about to activate. She takes a little damage, as you can see...
...but because she has a cold-resistance item equipped, she can continue to move at normal speed.
Here she has no cold resistance. When she moves over an active vent, you can see the plume of ice crystals trailing out behind her, and her speed is slowed to a walk.
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