The story "A Rose for Emily" is about the inability to accept and go through change to the largest extent.
The main character, Emily, is a memory of her former self: A woman who stubbornly refuses to pay the taxes that the new town Coronel requires, one whose father's dominance prevented her to develop as a woman and as a person, and as a former member of a privileged aristocratic Southern family who cannot accept that her former world of commodity is gone.
She is unable to accept death, and nearly did not give up her father's body for burial. She refuses to change her lifestyle to the point of keeping a black servant (as a form of slave) in the house. She totally snapped when her boyfriend Homer Barron attempted to leave her, and she went and poisoned him and lived with his corpse until her death.
Her home and her appearance is described as "an eyesore among eyesores" and the detrimental state of what once was a stately Southern mansion is also visible in the detrimental state of Emily, who still being pudgy, old, and strange was able to maintain the air of arrogance and self grandiosity that was once her right to have.
For these reasons, "A Rose for Emily" is in its entirety showing the many ways that resistance to change can manifest and the consequences that come as a result of it.
William Faulkner's short story "A Rose for Emily" can certainly be read as being all about the (in)ability to change. Just about everything has changed on poor Emily: the neighborhood is no longer grand, mailboxes have gone up, taxmen are knocking on the door, fashions have changed (e.g. china painting is no longer popular), and so on. Emily is not willing or able to keep up with these changes; her response is to attempt to freeze everything the way it was for her decades ago. Her dealings with Homer Barron can be viewed as being connected to her attempt to resist change.
If I recall correctly, her father's timepiece is a prominent part of Emily's wardrobe when the reader is first introduced to her. Somehow that detail strikes me as relevant to a discussion of not keeping up with the times.
In a broader sense, the story is frequently read as a reflection, perhaps colored by more than a hint of nostalgia, of the dramatic social changes in the South in the early decades of the twentieth century.