Yes, running a computer 24x7 can cost money. The only thing you can do, is utilize WOL, or start using stand-by settings, that it will wake up when it senses the HDMI port asking for a signal, then have it shut down after 30 min's or such, when it no longer senses the signal on the hdmi port.
There is no way around it, unless you start looking at habits and figuring how to change them to save money. Our household on average uses at the most 600kwh a month, and it costs us at the most about $75 for electric. Our average bill has been with Sewer, water, electric $106. I do not really worry about it, due to technology has become a everyday habit with most households.
I use a Ted5000 to keep track of what is using what, and can pretty much nail the bill down to the use per month for our household. In the middle of the night, it is funny to see our house only using one cent a hour, the most four cents per hour in the middle of the night.