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With the kind of money you want to spend, you can replace the graphic card too, you'll get better performance with a 1070 and an I5 than with an I7 and your old 970.
One chip runs cooler than another.
Not necessarily true. The temperature a device runs at is a function of the energy input and the heat removal.
You can make a Ryzen run hot. You can make an Intel run hot. Heck, you can make a ARM cpu that draws 1W run hot.
If your chip is running cool or hot, it's not because it's a good (or bad) chip, it's because of the heat sink and cooling you have selected to pair with it.
What you should be comparing are TDPs. That is a loose representation of how much energy a chip is going to generate as waste heat. It's debatable how manufacturers come up with those numbers, and how strictly those numbers are enforced, and under what conditions those numbers are generated, but it's a good, relatively standard metric that's used across the industry - not just CPUs.
If you have two chips, both rated at, say 100W TDP, and you put the exact same HSF on both: you would expect both to run at close to the same temps under the same conditions.
If you have two chips, one rated at 72W and one rated at 145W, you would expect one of two things:
Either the 145W chip will run hotter under the same HSF and operating conditions --
Or the 145W chip will require a larger HSF setup to run at the same temperature under the same operating conditions.
You could easily create a situation where the 100W chip runs much hotter than the 145W chip, just by adjusting the HSF that you use in the setup, or the ambient air temp, or the case ventilation, or any number of other variables. Hence, my pet peeve.
It would be fair to make the statement that Ryzen runs cooler with the stock HSF included with either CPU, as most high end Ryzens come with a really nice HSF, and high end Intel chips don't even bother with a HSF (and the ones that do include it is still laughable at best). But almost no one in enthusiast circles runs with stock HSFs, because aftermarkets are almost always better, even those for purchased very little money.