There are a number of possible reasons I can think of, ranging from resolution difference to overlays. I'll do a proper post/edit when I get out of bed and jump on my PC.
EDIT: Forgot about this post.
First thing's first, as Zetta mentioned below, resolution change. Normally when you first load a game, it will either auto-detect your native resolution, auto-adjust to your current resolution, or load onto a "safe" resolution (like 800x600 for Valve games) then let you change the settings yourself. Very often with fullscreen games, the process of switching resolution from normal to the "safe" resolution will cause your secondary monitor to black out (it's secondary after all) as it deals with the first monitor. Sometimes this doesn't get resolved until you alt-tab out (which breaks some games) or just close the game, since getting out of the game allows the desktop to restore normal resolutions and your secondary monitor.
A second possibility is if it is trying to force your secondary monitor into the first monitor (if the game coding clashes with multi-display systems) thinking that the two monitors are a shared single display. That or it will just ignore the second monitor to direct available resources to the primary monitor. It's also possible (if the secondary is smaller than the primary) that it will force the secondary monitor "out of range", i.e. trying to make it display a resolution it cannot handle (ever tried making a 15" monitor display over 1900x1080?).
As for overlays, some games use that for their cinematics. Overlays tend to black out anything except the cinematic, which means both displays go black then the cinematic displays on the primary monitor. After that, it just never removes the overlay from the secondary monitor until you exit the game. I've seen this happen a lot (probably because I've always used 2 monitors).
Then there's more abstract possibilities like your system running out of vRAM to keep refreshing both monitors, or driver conflicts, or other random stuff like that.
tl:dr, test it out on other games if it keeps happening, then look up forums for specific advice. I've yet to play Arkham Asylum so I can only give general advice.
