I understand from an email received from a Cox representative that the "Unmute" reset when a watchtv.cox.com channel is selected was an intentional design choice.
"Regarding the mute question that you have, it appears that this is the way it's designed. Since you have to actually close the window of the channel that you're changing from it cancels any audio options you had selected for that channel, as you know. When you open the new window for the channel you're changing to it's goes back to the standard, which is un-muted". - Cox Social Media Support Specialist
This explanation is illogical. Only one channel may be viewed at a time. No benefit is provided by assigning the "Mute" or "Unmute" option to a specific channel, particularly since it's going to be cancelled when that channel is closed anyway. Audio volume level applies to all channels and is retained until changed by the user or is reset to full volume the next time the Contour website is launched. The mute option should also be made to apply to all channels and only be automatically reset when the website is launched.
I suspect the choice to "Unmute" audio when changing channels may have been made to reduce the number of erroneous complaints Cox receives from customers that their audio isn't working when it's only because the audio has been muted. If so, a far less intrusive solution would have been to "Unmute" audio each time the Contour website is launched, similar to how volume level is reset, and then allow users to control it.