Does capillary action lower water temperature?

Capillary action is due to the "surface tension" (cohesion within the liquid). Surface tension is the energy per unit area of the surface of the liquid. The molecules of water inside the liquid are nicely sitting in a potential "hole" induced by the adjacent molecules.

However, the molecules of water that are on the surface lack the attractive force to the now non-existent neighbors on the side where the liquid no longer exists. Because attractive forces are related to a negative potential energy, the lack of them leads to a positive potential energy.

Consequently, water will contain some extra positive energy called the "surface tension". It's a form of potential energy but it can only be interpreted in this way at the molecular level. As hinted in the previous paragraph, the surface tension is a contribution to the energy that is proportional to the surface. That's why the surface tension causes water to take the shape of spherical droplets, among many other things, because the sphere minimizes the surface among shapes of the same volume.