When current through a zener diode increases by a factor of 2 by what factor does the voltage across the zener diode increase?

If the zener diode is in zener breakdown the voltage across the zener diode remains constant regardless of current (for the ideal zener diode).

Real zener diodes have parasitic resistance that causes the voltage across the zener diode to increase slightly with increased current, but due to temperature dependant variations in this parasitic resistance as well as temperature dependant variations in the zener breakdown voltage, this change in voltage in real zener diodes cannot be described by a simple linear factor.