Disclaimer: The purpose of the podspec's platform attribute is to make sure the library isn't installed with a version of the OS it has not been tested on by its maintainers. That being said, in your Podfile
you can simple change the platform requirements to what you want to emulate for example,
platform :ios, '6.0'
Obviously that is for all your pods not just a single one but you can see why that feature doesn't exist. In the newer versions of CocoaPods you actually don't need that line at all and it will detect your target version from your project, obviously since you're trying to use code that's not meant for the version you're using that wouldn't help you but it's typically quite useful.
Edit:
Alternatively you can edit the spec's source directly. In this case open ~/.cocoapods/master/ARChromeActivity/1.0.0/ARChromeActivity.podspec
in some editor and change:
s.platform = :ios, '6.0'
to
s.platform = :ios, '5.0'
Then run pod install