Maybe you could argue it cost some electricity, but... In reality, it meant my computer, which runs 24/7 pulling ~185W, was running at ~300W for 56 hours... Thusly.. 300 - 185 = 115W * 56H = 6.44kWh @ $0.13 per kWh = $0.85 + tax.
So... Yes, it was very much worth $0.85 to make my wife happy.
You would want to add the cost of your network+hardware depreciating over the timeframe, and you probably can't just ignore the first 185W since if you are Anthropic it doesn't seem likely that the idle power draw would be needed if they weren't expecting to serve AI traffic.
So, let's say $0.02 per hour ($1/50 roughly). That's about $15 per month per user. Let's call it $10 per month per user since users aren't constantly hammering the service. To support a big sales and marketing engine, you would like to be selling subscriptions for $100+ per month. I'm just not sure people are prepared to pay that for AI in its current form.