An application contacts an Internet host on each start of it. It seems to take just an instant. I don't know if HTTP, plain TCP or UDP or what is used but tend to suppose HTTP. What a tool can I use to track this connection?
Watch all network traffic
In a terminal, as root, run
and watch the packets fly. If you have other network activity, filter that out. For example, to show everything except incoming and outgoing SSH traffic:
Watch what the application is doing