Is there a chemical reaction that speeds up when the temprature gets lower?
That's a good question.

A reaction slowing down on a rise in temperature seems counter intuitive but it's not so always.

The Arrhenius equation works for only a few limited elementary reactions and the transition state theory is a much better explanation of rate laws and their variation with temperature.

The dependence on temperature of the reaction rate cannot always be analysed purely using the Arrhenius equation. It can be better understood by knowing the mechanism of the reaction.

For instance
$$\ce{2NO + O2-&gt; 2NO2}$$                   

Has its first step as 
$$\ce{2NO &lt;=&gt; N2O2}$$ 

And the second step as:
$$\ce{N2O2 + O2-&gt;2NO2}$$

The first step is exothermic in nature.
And the equilibrium shifts to the left on increasing temperature based on the Van't Hoff equation of gaseous equilibrium.

So the effective rate constant of the overall reaction decreases on increasing the temperature.

Thus, we see a reaction whose rate slows down with the rise in temperature.

There are in fact many such reactions.

To give another example, I could take an enzymatic reaction in a living cell and show that the rate of reaction falls down drastically with a rise in temperature. Ever wondered why your body rises its temperature during a fever? The aim there is to slow down the reactions in the cells of the invading germ and hence kill it.

From these examples, one could conclude that the famous Arrhenius equation works in only a few limited(and ideal) cases.