Is this an example of pathetic fallacy: "Time went by relentlessly and it was Saturday again."?
As written, no. But I think changes can be made to make it so. See below for more:
Pathetic fallacy is when a writer gives inanimate objects of nature human qualities and emotions in order to reflect a mood. It's a type of personification (which is the giving of human qualities and emotions to any sort of non-human object).
Pathetic fallacy is often used to set a mood with weather. For instance, I can write:
The sombre clouds drearily hung about the spires of the ancient castle.
The clouds in the sentence help set the mood for what we can expect in the castle - must, dust, mold, rot, etc.
And so to the question at hand - is "Time went by relentlessly" a form of pathetic fallacy? Is a mood set with the sentence? Are human qualities given to time? To my eye, the answer is no - the key here is the word "relentlessly" - while the attempt is to have that word create the mood and do the associations, "relentlessly" simply means "fast".
Can we change the sentence to make it pathetic fallacy? I think yes.
Time relentlessly rushed passed. And it was Saturday again.
Does this meet the definition - is a mood set? I think so - there's agitation, speed, things happening much faster than the narrator wants. Is there personification given to an element of nature? It's arguable if time is a part of nature, but if it can be, then we've given time the ability to rush in a determined way.