Gregory (Formal Logic, p. 244) states that one of the requirements for the universal generalization is that the variable $a$ does not occur in the generalized formula $\forall x P(x)$. Gregory also says that if $a$ does not occur in $\forall x P(x)$, then all occurrences of $a$ in the formula to be generalized must be replaced by the bound variable $x$ in $\forall x P(x)$. Example: $P(a, a)$ cannot be generalized to $\forall x P(x, a)$, since $a$ appears in $\forall x P(x,a)$ (not all occurrences of $a$ were substituted by $x$).
However, I came up with a possible counterexample: $P(a, a)$ cannot be generalized to $\forall x \forall y P(x, y)$. Obviously, $a$ does not occur in $\forall x \forall y P(x, y)$. The example makes me think that the fact that $a$ does not occur in the generalized formula can't stand alone as a requirement for the universal generalization.
Gentzen (Investigations into logical deduction, p. 78) does not refer to the occurrence of $a$ in $\forall x P(x)$ as a restriction for the universal generalization. Instead, the author cites the arbitrariness of $a$ and that "every occurrence of $a$ in $P(a)$ must be replaced by an $x$ in $\forall x P(x)$". To make this restriction applicable to our hypothetical counterexample, I assume we should interpret that $a$ must be replaced by an unique bound variable $x$ in $\forall x P(x)$. Is that so?
You are correct. The reason that this is not stated out loud is because your potential counterexample is really two generalizations that you've done in one step. Formally, this is not allowed.
Say you have the formula $P(a,a)$ and you want to turn it into $\forall x . \forall y . P(x,y)$. Then we have to first generalize to $y$, then to $x$.
So to construct your counterexample, we are really doing this: $P(a,a) \rightsquigarrow \forall y. P(a,y) \rightsquigarrow \forall x . \forall y . P(x,y)$
Of course, the above sequence of generalizations is NOT allowed, because we did not generalize all of the occurrences of $a$ when we added the $\forall y$.
You'll find that if you break down generalizations to one quantifier at a time, the only thing that can go wrong is accidentally forgetting to rename every occurrance of the variable being quantified. Of course, this is exactly what the rule of universal generalization says.
I hope this helps ^_^