I would have to think that the space race is a thing of the past. With the economies of the two leading nations the way that they currently are I would think either government would have a difficult time justifying the use of money in this way.
Just a small clarification of terms here. The term “cold war” generally speaking refers to the use of threats and intimidation to try to gain or maintain control in a particular circumstance. Such as the Soviet Union and the the United States in the building up of their arsenals. A hot war is actually one in which guns, weapons, and the such like are used. A cold war in space is possible, but world nations seem to be working cooperatively for the time being.
I think the closest thing to weapons in space was the "Star Wars" defense system which was started under President Ronald Reagan. The system was intended to defend against Soviet ballistic missiles if fired at the United States.
Knowing the rudiments of lasers, and the facts of satellites, I cannot help but wonder if in fact opponents during the Cold War didn't routinely (and quietly) "knock out" each other's equipment from time to time. Factually, both sides used spy satellites; had the Cold War persisted it only would have been a matter of time when manned and armed space vehicles entered space.
This current age cannot exist without computers, and computers grew up during and because of the Cold War. Current astronomy relies upon them, and they have opened our eyes to the complexity and wonder of the Universe. They've also spotted some troubles, as noted. For the first time, it's possible to monitor such troubles long in advance and prepare. Perhaps we won't go quite the way of the dinosaurs.
There was an agreement between the Soviets and the US during the Cold War not to weaponize space. Both sides wisely believed that were that to happen, it would be almost impossible to avoid another expensive and dangerous arms race, and World War III was likely to be the result. So, as far as we know (though it may have been done secretly), no actual weapons systems were deployed.
The SDI "Star Wars" program to defend against ballistic missile attack was explored, funded, and researched, but never deployed. Most scientists think it unfeasible.
There has not been any use of any weapon against meteors (actually asteroids are what one might use weapons against). There are proposals that we should be monitoring asteroids that might hit Earth, but nothing has been seriously done about this.
It is not likely that we will have an "Armageddon" (film) type situation any time soon. Our missile defense system is still a long way to go. I remember a general saying that we had 100% with the Patriot missile defense system when in reality it was more like 10%-25% accurate.
"Cold War in Space" and "Conquest of Space" are very high sounding phrases that convey little meaningful information. Impact of massive meteors are real threat to the earth and the humanity, just as other natural disasters like earthquake, and volcanoes are. But actions to prevent or control natural disasters does not qualify as any kind of war or conquest, except as a figure of expression.
When we speak of outer space which encompasses the entire universe with its billions of galaxies, the comparative importance an power of earth is less than the importance and power of a speck of dust in comparison to the earth. avoiding or controlling the impact of meteors is just too insignificant to be labeled as "conquest of space".
UN prepares for a situation when, not for long,the Earth will be hit by an asteroid. Asteroid Apophis will be dangerously close to Earth, in 2036. Specifically, there are 45,000 to one chances to hit the Earth, the situation in which any trace of life will disappear from the region where it will fall. It all depends on the angle under which the asteroid enters the atmosphere and especially what kind of rock is formed. But the UN can not ignore this scenario, no matter how big or small it is likely to occur. So it organized a "mission for asteroid deflection", a complex group composed of astronauts, engineers and scientists who are seeking to deflect the celestial body. So far, the best option seems to be diversion of Apophis with a spacecraft that would use gravity to alter the asteroid. Apophis was discovered in December 2004 with the Large Synoptic Survey Telescope Telescope (LSST). The asteroid with a diameter of one kilometer has a 323 days-revolution around the sun. In 2036, his path, practically will cross with that of the Earth . This mission will cost about 300 million dollars, acording to CNN.