I was thinking about temperature, and had a thought. Temperature is proportional to the average temperature per particle. If you split a particle, you have twice as many particles but only the same amount of energy. So, each half would only have (on average) half of the original energy.
The first example I though of would be salt dissolving in water, which I looked up and it is slightly endothermic. Unfortunately there are a lot of factors at play that both increase and decrease the energy during that reaction so it isn't exactly conclusive. Plus, if this were correct you would have half of the temperature in kelvin, which is an enormous temperature drop which you definitely do not see in dissolving salt.
Did I make a mistake or does a particle splitting actually decrease the temperature? If so, why does salt dissolving not cause a massive temperature drop?