It certainly does. There is for example a measurable energy cost for having a blinking cursor in a terminal, and there have been huge flame wars about efforts to move to non-blinking cursors.
The compromise for GNOME Terminal is that the cursor will stop blinking after a terminal has been idle for ten seconds.
Why not similarly disable seconds display after some specifiable time has elapsed since the last key or mouse event? The decimal digits could be replaced with a greyed-out ":--".
It's going to take power, no matter the operating system. What matter is how much power it takes. On most desktop environments and widgets, it's probably negligible.
The compromise for GNOME Terminal is that the cursor will stop blinking after a terminal has been idle for ten seconds.