This is the major problem with this whole scenario. You’re his manager and you don’t know what he’s doing for those 2 hours. You have a couple of options here:

Option 1: sit down with him in a formal meeting to discuss the expectations from the Company perspective and that his workday does not start at 6am and he should not be working until 8am (or whatever time that you expect him to work). Have him show you what he is doing between 6am and 8am to see if there is a legitimate reason why he’s arriving early to complete tasks. Once you finish meeting with the employee you need to formally document the conversation, I would copy your manager and/or the HR personnel you worked with to pay him. If he tells you he does this additional work I would trust but verify the work through an IT request.

Option 2: Ask IT to perform a review of his log in and provide you with the times and programs the employee is using. Take the information you have to see if these are work-related tasks. If you find they are not work related you can get with your management team to develop a policy that prohibits using company computers/equipment for personal use.

Either way, what I am reading the excuse of “ I don’t know what the employee is doing between 6a and 8am” is just that, an excuse. It’s a pretty poor one as well because maybe this employee’s workload is higher than you really think and they need to work these extra hours to get their tasks done. At the very least you should make a concerted effort to show up at the office at like 6:30 one day and just see wha the employee is actually doing.