The whole purpose of Adobe AIR is build desktop applications that may or may not connect to the internet. With Klok, the only time you need an internet connection is when you have it check for updates automatically (this feature is new in 1.5). Other than that, everything is done locally on your system. The datafile is stored to your filesystem.
In version 2, I will be introducing the ability for Klok to integrate with 3rd party or homegrown web-based time tracking systems. For you to use that ability, you would obviously need an internet connection to synchronize. However, even when integrated with another web-based system, the plan is for Klok to still work even if there is no internet connection and then when a connection is available sync up with the system