US Global Entry is now available for Australia Citizens, however in "phase one" it's only available for people that have visited the US 5 or more times in the past 12 months.

https://www.cbp.gov/travel/trusted-t...alian-citizens

Enrollment is via the standard GE website, but apparently the application is then forwarded to the Australian Government for a background check before then going through the normal GE entry process. There's no need to provide any details of prior entries to the US, but at the final step in additional to making you certify you meet the requirements of having been 5 times in the past 12 months, it states that any applications that don't meet that requirement will be help until Phase Two which implies that they are checking.

I've applied, and my application is in "Pending Review/Waiting for Conditional Approval" state - I'll update if/when that changes.