I can't recall reading a single cozy that wasn't written in past tense. It is the most common tense to write in. Most people prefer reading past tense to present tense--myself included. And past perfect or future tense would just be annoying.
But could a cozy work in present tense? One of the arguments for using present tense is that is makes the action more immediate. Everything is happening NOW. The reader is experiencing everything as the protagonist does.
Yeah, well, that's one of my problems with reading present tense. I know I'm not in Cute Little Town, Arkansas, walking into a bagel shop. Present tense throws me out of the story and unsuspends my disbelief.
So, how about you? Does present tense bother you? Do you think it would work for a cozy?
Inquiring minds want to know.