In short this is the fault of GOG, not Microsoft. Installers and other applications needing access elevation are supposed to incorporate metadata to tell Windows that they need administrative intervention (this can be implemented in a variety of ways, but the outcome is the same). Vista and 7 can detect this need in a fair percentage of non-compliant software via heuristic analysis, but this doesn't work for the GOG installer format. This change is immediately visible; software that incorporates the appropriate metadata or is heuristically detected will display a UAC shield in the corner of the icon. If the shield isn't there, UAC will kick in later on or sometimes not at all.
Because GOG isn't supplying this information and Windows can't detect it the installation process is disrupted, which you are seeing in the form of two integrity checks. If you force the installer to need access elevation (most easily achieved by right-clicking and choosing "Run as administrator") you will receive a UAC prompt immediately, and the integrity check should only happen once.

Arkose is correct. This is a failure of the GOG packaging format, not Windows. As soon as Vista/Win7 detect the installer trying to access parts of the hard drive that are labelled as protected sections, it asks if you want to elevate the program's privileges. But the GOG installer only does this after the file check is mostly or fully complete, which is why you end up running the check twice.
Really, it's not difficult to press the Skip button, or to tell it to Run As Administrator before you even start.

I didn't expect this to be an issue for me, since I told my UAC to go fuck itself.

If you turn off UAC, it will not do this. I have mine off and have never had a problem. User Acount Control is a sercurity feature that does more harm than help. I would recommend turning it off.

Like I said, I turned mine off (about five minutes after installing Vista), and I still get the two integrity checks.

It could be the virus scanner that you use. I changed alot of things on vista to get it to work right for me, a few dozen things, so I don't know why it doesn't happen to me...

Not true. UAC is a security feature that is more intrusive than helpful, at least on Vista. But it's a good security feature to have. Win7 is more intelligent about the UAC implementation, and I find it to be both functional and secure.

The Vista version is stupid, at least:
"Uh-oh, I've been asked to open Notepad, and as we all know, that's a terribly dangerous program to run on your computer. I don't think I dare! Could you please hold my hand and tell me it's okay to open Notepad? Here, just fill out this verification form in triplicate. Mail one to Microsoft, mail one to the NSA and with the last one, make a funny hat so you can look like the dumbass we obviously take you for".

When I can't run a program because of UAC, it is harmful and it did that to more than one program of mine when I first got vista. Of my knowledge it NEVER stopped a virus from running on my PC. Thus it is more harmful than helpful.