Would cooling of a battery using cooler affect its life time?
Let's see what I know. Not really sure whether this is a terrific answer to your question, but it's what I have.

Through knowledge of using laptops, I know that lithium-ion batteries will decrease in performance when overheated. This probably applies to most redox reaction batteries for the reason that the reaction proceeds more quickly when temperature is increased, and as a consequence, kinetic energy of the molecules are increased.

Increasing the temperature of the battery also increases internal resistivity, which is somewhat more of a physics question. The electrons collide with the material that they are flowing through much more frequently in a higher temperature situation as opposed to a lower temperature situation. This doesn't mean, however, that you can dip your phone in LN2 and have it work forever. There's obvious physical limitations to everything. 

What you're doing to your phone (a.k.a. cooling it) shouldn't have a problem at all on your phone's performance, mainly because you're doing to it what others do to their laptops - placing it on a cooling pad.