The color temperature represents the color of light emitting from an ideal black body, when heated to said temperature in kelvin, and how an incandescent bulb emits light is close to a black body.
Does this mean a 5000 K incandescent bulb has higher filament temperature when powered than a 3000 K? Does a 5000 K incandescent bulb use more power than a 3000 K? (EDIT: under the same brightness in lumens)? Does a 5000 K incandescent bulb produce less brightness than a 3000 K when at the same wattage?