Linux

First-person survival game where you're stranded naked and helpless on an island full of dinosaurs.

Interstellar Marines bills itself as an old school tactial FPS, and just recently added a solo and coop player campaign mode called the Neurogen Incident.

Proven Lands is a sci fi survival sandbox game with release planned in 2015 for Windows, Mac OSX, Linux, iOS and Android. Solo, multiplayer and cooperative modes are included, and it is currently being both Kickstarted and Greenlighted. Check out the video, this one may be worth looking into next year!

The collection is actually the first game in the AI War series, Fleet Command, and five expansions released between 2010 and 2013. The developer, Arcen Games, is working on a new entry in the series due out later this year, called The Last Federation. All feature competitive and cooperative play against human or AI players. It runs on Windows, Linux and Mac OSX, and supports cross-platform multiplayer.
Steam has just put up for sale the entire AI War Collection: six titles for less than five bucks!

This free, open-source empire-building game lets  you lead your tribe from the stone age to the space age. Now there's an HTML version playable in browsers, even on mobile devices! Features network and play-by-email options as well, and supports Windows, Linux and OSX with cross-platform play. Currently the OSX version is out of sync with the latest stable version (latest Mac version 2.3.0, latest stable for Win/Linux 2.3.5).

The successor to the Quake engine-based Nexuiz, Xonotic is a fast-paced multiplayer shooter, similar to Unreal Tournament or Quake 3 Arena. Features FFA and team play.

According to its sourceforge page, Triple A is "a free online turn based strategy game and board game engine, similar to such board games as Axis & Allies or Risk." While it doesn't appear to have a discrete cooperative play mode, these kinds of games usually feature the ability for players to create informal (and usually temporary) alliances among themselves, so I'm including it here. Thanks to a former IMG writer who tipped us off to this gem!

Built from the ground up for cooperative play and supporting OSX, Linux, and Windows, Left 4 Dead 2 definitely belongs on this list. And for a short time during the holidays in 2013, Steam offers the game for free!

This heist game isn't quite GTA but it is a cooperative bonanza: it works on Windows, Linux and OSX and supports both network and local cooperative play as well as cross-platform multiplayer. PC gamer in specific called out its excellent cooperative play mode.
Plus, it's on sale on Steam right now!
In fact, I think Blackstar and I are going to try this out at some point. I may even have some extra codes if there's interest...

A mix of roguelike and classic hack and slash 2d dungeon crawling gameplay, Hammerwatch is the complete package that features single player, cooperative multiplayer online or on a LAN, as well as a level editor-- and it runs on Windows, OSX and Linux and allows for cross-platform play. 
Title is available both on GOG and Steam. Steam has a 4-player pack for $24, but GOG sells each individual copy for $4 right now, so that seems like a better deal.
DRM-free!

Released on Early Access in December 2013, this first person action adventure RPG  features cross-platform multiplayer on Windows, OSX and Linux. The game looks to be pretty early in development, but seems to be snaring enough people to invest the hours in playing it.

Out now on Steam on Early Access, Entropy is listed as a massively online action strategy RPG. Sounds a bit ambitious, no? And to boot, cross-platform multiplayer on Windows, OSX and Linux. (I'm wondering if the phrase "Unity Engine" is in there somewhere... it does look like the studio has used it in the past.)
Here's what the devs say:

Markus "Notch" Persson created this block-based game in 2009 using Java. Years later, versions exist for PC, OSX, iOS, Android, and Xbox, with a PlayStation version in the works. 

OpenRA is a modern crossplatform engine for the Command and Conquer series of RTS games. Supported games include the original C&C, Red Alert, and Dune 2000. This engine requires the original game data files to work. They were released as freeware but appear to be no longer available from EA following the announcement and subsequent cancellation of a new F2P C&C Game. The original C&C was first released in 1995.

0 A. D. is a real-time strategy game of ancient warfare where players build and develop their own city-state to compete with other human or computer-controlled city-states.

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