You could see on the left side with the small metal hose that it’s quite damaged. It was busted in and was leaking something yesterday.
My 2006 s40 has been having a few problems. Leaking antifreeze and overheating. I just replaced the thermostat and checked under the car and found this hose was pretty busted, so I’m looking to replace it. Anybody know what this hose is called? Thanks!

 

looks like your oil cooler line. I'd check out a diagram from a Volvo dealer web site to see the proper part numbers for both of the oil cooler lines.

 

That's your transmission oil cooler line - a dealer should be able to help you find the right part (I didn't have any luck with a quick online search). You definitely want to replace that before your tranny is damaged from low fluid levels (good time to do a drain-and-fill while you're replacing that line, since it's kinda necessary anyway.