Do various soaps work better when dissolved/used in different water temperatures? My wife and I have had an ongoing argument about various cleaning agents, and I wanted to get a Chemists' view on it.

When we are doing laundry, she will generally clean using the cold water setting and a standard amount of detergent.

If I do laundry, I will use the warm water setting with a standard amount of detergent.

Either way, the clothes come out (seemingly) clean. Using cold water saves energy from the heating process, which has been her excuse for doing such a dastardly thing. Her argument against using warm water is that it will damage/shrink the cotton. While I don't disagree on that point, I am curious if the use of warm water causes the detergent to work better or not.

In addition, there are various soaps that work best in cold water.

What is the actual mechanism at work here? Why do some soaps work better in cold (or hot) temperatures?

To make things a bit better, let's remove some more variables. We use this detergent:

Tide Pods

Unfortunately I can't find information on the normal temperatures those machines operate at.

Also who is right? (I bet it's her)
To start, soaps and detergents have two different chemical makeups, meaning that they act different under different conditions.

For general soaps, here is an explanation of how it works:
http://chemistry.about.com/od/cleanerchemistry/a/how-soap-cleans.htm

As a general chemistry note, as the temperature of a solution increases, the ease of a chemical to change (whether it is to react or change shape) increases.  This means that colder temperatures will allow a chemical to remain in the same state.  If the soap depends on a specific conformation/shape to act then colder temperatures may be preferable.  However, if a reaction needs to occur in order for the soap to act some heat may be required.

For Tide Pods, it states that it "dissolves quickly in hot &amp; cold."
http://www.tide.com/en-US/product/tide-laundry-detergent-pacs.jspx

The chemical makeup of Tide Pods is composed primarily of salt compounds and enzymes.  Both salts and enzymes tend to operate better at higher temperatures.  So as long as you don't pass the denaturing temperature of the enzyme, higher temperatures would be preferred.