Why does New Zealand and cold countries have volcanoes?
Surface temperatures on Earth are a consequence of solar heating and radiative cooling - in other words, the heat energy which drives climate comes from sunlight falling on the Earth "now".

On the other hand, the heat energy driving volcanism is left over from the formation of the Earth (4.5 billion years ago), stored deep inside the planet, with some additional heat generated from the radioactive decay of elements contained in the rocks.

Deep within the Earth, the temperature is on the order of 5000-7000&nbsp;K. It's obviously much cooler at the surface. The high temperatures cause the rock and metal of which the Earth is composed to melt. The temperature gradient results in a slight density gradient, which drives convection (flow of hot material up to the surface). When that hot material breaks through the surface, you have a volcano.

Consider that the hottest places on Earth (in terms of climate) don't have surface temperatures anywhere close to what it takes to melt rock (maybe 50&nbsp;°C compared to 1000s of °C). Heat can only move from where the temperature is higher to where it is lower. Less heat falls on the planet from the Sun than the Earth radiates away into space; that difference is heat finding its way up to the surface from within. So, the heat driving volcanoes cannot possibly be coming from sunlight striking the Earth; it can only come from heat stored inside the planet from long ago. Consider also that the difference in temperature between the hottest places on the Earth and the coldest (in terms of climate) differ by little more than 100&nbsp;°C. That's a tiny fraction of the temperature difference between molten magma and even the hottest desert. Bottom line: climate has no effect on volcanic activity.

Since the Earth is losing more heat to space than is falling on it from the Sun, and that heat is coming up from below, the core is gradually cooling. But there is so much heat inside the Earth and the rate is so slow that our Sun will die before Earth's core cools enough to solidify.