I have bought a cross-platform game over steam and installed it in Linux. Would I have to buy another key to play it on my Windows pc? or could I just install it on my windows pc?
I assume "cross-platform" refers to .
According to Steam support, you can just install it on windows.
What is Steam Play?
Steam Play allows you to purchase your games once and play anywhere. Whether you have puchased your Steam Play enabled game on a Mac or PC (both Windows and Linux), you will be able to play on the other platform free of charge.
The buy once, play everywhere part is the central part of Steam play.
If the game supports Steam Play with Linux, you should be able to install the game without rebuying it.
Otherwise, you'll most likely need to rebuy it or run it on a virtual Linux machine.