I have double feelings about this battery draining stuff right now. I found this answer on their FAQ field and it kinda says different things from the first quote I made: "The Globalhop SDK may be set up to work only while charging, so battery draining may be completely avoided. Even though if you chose to keep SDK working on battery, it is build so, that SDK does it’s job and goes off and doesn’t use any measurable battery resources. The SDK is working while device is idle only, so user will not meet any slow down effects. Also, SDK doesn’t process any large calculations, so it does not use much computing power." So on one hand, it can drain the device battery, but you can change the settings and avoid that.
From what they are giving out on their page, I think there are several things they could be doing, like cooperating with companies and testing page availability from various devices/locations, and so on. That's only my theory of why they would need to route some traffic but as they're saying it's GDPR safe, I don't think there would be some shady things going on your device background.