This obviously has considerable political implications since it will inevitably be intertwined with climate change. So it's not merely a scientific discussion.
Will be kind of fun if the term becomes institutionalized, then some new groups propose the "Romanocene", the "Coloniocene", the "Industriocene", the "Americanocene" etc etc.
The political implications may be the strongest argument against it. Science should strive for neutrality, to present facts without getting tangled up in politics.
On the other hand, now that the previous commenter mentioned that, I can see an objective reason to be against the term 'Anthropocene' regardless of whether the epoch itself is accepted.
'Anthropocene' implies that it is humanity that is the driving force of the purported geological changes, but humanity's ability to cause such an impact is itself a blip in the biological history of our species. I do see that some proposals as to when the epoch started go as far back as the Neolithic revolution, but I very much doubt that a hypothetical world in which humanity had never progressed past subsistence farming and husbandry could have an effect on the geological scale. Plus, humanity still has more history before the Neolithic than after it.
So, taking the more mainstream arguments stating that a new geological epoch started in modern times, be it with the Industrial Revolution, the Atomic Era, or the 1960s, the fact that this epoch is Anthropic is not necessary and sufficient to describe it.
Instead, I would argue that the defining characteristic of this epoch would be Industrialisation, hence 'Industrocene' or, forgive my Greek, 'Viomechanocene' would be much more descriptive and more objective names.
Will be kind of fun if the term becomes institutionalized, then some new groups propose the "Romanocene", the "Coloniocene", the "Industriocene", the "Americanocene" etc etc.