A big issue I have with act utilitarianism is the way it self-destructs pragmatically.

It looks like better consequences will arise if we teach a form of deontology, reward or punish people who (respectively) follow or break the moral rules, call it "right" to follow the rules and "wrong" to break them etc. So a true act consequentialist will encourage everyone to become a deontologist (and to the extent others copy him, will act like a deontologist). "Rule utilitarianism" seems immune from this problem, though arguably rule utilitarianism is a form of deontology; it just has an underlying rationale for selecting a particular set of rules (i.e. the optimal moral code).

A different objection is that it is simply too demanding: the best way for me to maximize utility is to give nearly all my money to humanitarian charities, so why aren't I doing that? (Answer, because my personal utility function has very weak correlation with a global additive or average utility function; though it does seem to have a strongly weighted component towards me personally following deontological rules. Funny that.)