Unfortunately the voltage has nothing to do with the heat. The HID units produce far more heat than standard xenon or halogen bulbs. The reason for the ballast is to limit the current drawn and thus stop these bulbs from melting themselves. Without ballasts the bulbs would just drawn more and more current, getting hotter and brighter, and finally meltdown the wiring loom and the headlamp surrounds. Obviously fuses may blow to stop all this from happening also but how many people stick "the next one up" when replacing blown fuses? It only takes two different people to stick bigger fuses in and you are running your headlights on 40amp fuses.
However, I digress. There is a history of these HID conversion units, especially the early ones, running so much hotter than the original bulbs that they have melted the surrounds.
Do you know the name of the supplier of your kit? I do not want to buy a cheapo kit with the chance of meltdown.
I asked if you had been on a long night drive because it is prolonged use that will melt the surrounds. Short trips to the local shops probably will not cause a problem. The last thing you want is to find your headlight conversion is ****e as it fails at midnight on a long journey home.