You got it!
This is definitely the case for a saturated vapor style burners (where just a seemingly little extra temperature means quite significant pressure rise, but mainly it means the vapor density increase), the result is then the mainly the arc holding voltage (so the level after the zero cross reignition spike) rises, the reignition voltage then goes along.
but here the question was about an unsaturated vapor burner: There once all the dose gets evaporated, the vapor density remains remains constant (for nitpickers it slightly decreases as the burner expands, but that is so little it has no impact at all), the vapor pressure rises only very slightly (compare to the saturated vapor concept; it just follows gas thermodynamic equations, so pressure proportional to an absolute temperature).
So because the density can not rise any further and the pressure change is way less steep, I have some doubts the pressure itself is really the cause for EOL cycling (it would require so large temperature rise, the materials would melt and so the tube explode before the arc voltage really changes in any significant manner), I'm rather suspecting some other mechanism, obviously somewhat temperature dependent.
My guess is more in the direction hotter gas -> faster charge density decay -> longer time to "rebuild" the arc after current zero cross.
Or another plausible: The high temperature causes the partial pressure of the other materials (tungsten, quarz components,... - mainly when approaching the melting point; all evaporate to some extend, only normally the pressure is so low it does not influence anything) rises and so contaminates the internal atmosphere, causing less arc stability. Or some contamination gets released from those materials due to the overheated condition.
