I have the same issue here, but I doubt it is Cookie related. I have two Android phones. My personal one is a Samsung Galaxy II(SPH-D710) with Android 4.0.4. My work one is a Samsung Replenish(SPH-M580) with Android 4.0.4.
I started using AirDroid with my personal phone and it stayed connected all the time. Even if left alone for a couple of hours.
My work phone will drop the connection every few minutes, even if I am doing work on it. I've been cleaning up my address books on both phone, but I've had my work phone for a couple of years and I had been using it for both personal and work. I got tired of coworkers calling on the weekends and friends calling me at work, so I got a separate personal phone.
Any ways, I get about 10-12 contacts deleted and the connection on my work phone drops. I have to refresh the page and re-enter a new code.
In both cases I was connecting by the IP using my home WiFi.