Why did Mars’s core cool down

The reasons are:

1.  After the Big bang, the energy burnt out, the whole universe cooled down gradually :  The truth is, not only Mars was cooling down, even our Earth is cooling down too.

ref:  no. 1

2.  Obviously , Mars is further away from the Sun, and also it is  a lot smaller than the Earth, that is why it is cooling down more quickly.

3. Mars has a lot less green house gas .

ref:  no. 2

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ref:  no. 1

The cosmic background radiation is radiation left over from early development of the universe, and is a landmark proof of the Big Bang theory. Before the formation of stars and planets, the Universe was smaller, much hotter, and filled with a uniform glow from its white-hot fog of hydrogen plasma. As the universe expanded, both the plasma and the radiation filling it grew cooler. When the universe cooled and stable atoms could form, they eventually could no longer absorb the thermal radiation and the universe became transparent instead of being an opaque fog. The photons that from that time have been propagating ever since, growing fainter and less energetic. The CMBR has a thermal black body spectrum at a temperature of 2.725 K, so it peaks in the microwave range frequency of 160.2 Ghz(1.9 mm wavelength).

http://www.universetoday.com/79777/cosmic-background-radiation/

https://nightsky.jpl.nasa.gov/news-display.cfm?News_ID=631

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ref:  no. 2

scientists believe, Mars must have lost its most precious asset: its thick atmosphere of carbon dioxide. CO₂ in Mars's atmosphere is a greenhouse gas, just as it is in our own atmosphere. A thick blanket of CO₂ and other greenhouse gases would have provided the warmer temperatures and greater atmospheric pressure required to keep liquid water from freezing solid or boiling away.

http://science.nasa.gov/science-news/science-at-nasa/2009/06nov_maven/