It seems that when one part of the country experiences very high temperatures, other parts of the country experience cooler than normal temperatures. This can be explained by the path of the jet stream. In other words, ‘weather’.
What about climate change? If the average temperature of the Earth is increasing, will the increase in local climates in some areas result in cooler climates in other areas?
Yes, if the average position of the jet stream changes; for example, recent extreme winters have been linked to melting ice in the Arctic, which results in abnormal warmth where there is a lack of ice (since open water can’t go below freezing but ice-covered water can get very cold), which in turn affects the location of the jet stream (tends to make it wavier, meaning sharper dips south and potential for cold air outbreaks). This also includes new weather patterns not previously observed.
I’m pretty sure somebody is going to have a very interesting winter this year (although, this isn’t a guarantee, as in the U.S. last winter, while Europe had major cold outbreaks).
If you are referring to “zero-sum” worldwide climates, I asked a similar question a few years ago. I don’t think there was a definitive answer, one way or the other.
If the average temperature is increasing, then it’s not zero-sum. 
Temperature variations in the short term are meaningless with respect to global climate change. Not only can you have hot years and cold years, you can even have hot decades and cold decades, and all of that is still so short term that it’s meaningless. Heck, the Holocene Thermal Maximum lasted for thousands of years. What’s a decade or two compared to that?
On average, the Earth right now is getting warmer at about 1 deg C per century or so. Short term variations in that are not an indication of global warming (or global cooling).
Climate change and local climate are two completely different things. On average, the entire Earth is getting warmer. Locally, you’ll still have some places hotter than average and some places cooler than average. If the average is about 14 deg C and one place is 2 deg warmer and another place is 2 deg colder, then as the Earth heats to an average of 15 deg C you’d expect the first place to be 17 deg C and the second to be 13 deg C.
But it doesn’t have to go that way. There are all kinds of interconnected feedback systems in the Earth’s atmosphere. If you disturb them, you can end up with dramatically different patterns than what you see today. At one time there was very little difference in temperature between latitudes where New York is right now and where Florida is right now. The equator was habitable and the polar region had trees, not ice. At other times, the equatorial regions became so hot that almost all life there became extinct. Things could get very different as the world heats up.
