Most likely, Games requires the graphics card to be put to work which generates heat and in a laptop that's a match made in hell as most of them have poor to inadequate cooling which will result in the components hitting their temperature treshold pretty easy and forces the laptop to shut down to prevent damage.
If temperatures get to high in a desktop that's just a matter of opening the case, clean out the dust and if needed replace the thermal paste on the CPU and/or GPU. A laptop however again that's a nightmare unless you know your way around it. If there's a lot of dust in and around the fan entries and fan blades it might be worth a shot but I would suggest to see if there's a work around for that game of yours to be played on that desktop of yours.
Does that game have a forum? Perhaps its worth asking if anyone is having the same issue with their graphics card and the game not working and maybe knows a way to fix it.
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