= Solution
For independent internal rotational states, the rotational factor in the many-molecule <partition function> is $q_{\rm rot}^N$. The indistinguishability factor has already been included in the translational part and is not applied a second time to the internal factor. Thus
$$
A_{\rm rot}=-Nk_{\rm B}T\ln q_{\rm rot},\qquad
\frac{S_{\rm rot,m}}R=\ln q_{\rm rot}
+T\left(\frac{\partial\ln q_{\rm rot}}{\partial T}\right).
$$
A nonlinear <rigid rotor> has three rotational quadratic degrees of freedom; in the high-temperature limit, $q_{\rm rot}\propto T^{3/2}$. Therefore
$$
\boxed{S_{\rm rot,m}=R\left(\ln q_{\rm rot}+\frac32\right).}
$$
This is the <rotational entropy> contribution, with the appropriate <rotational symmetry number> already inside $q_{\rm rot}$. It has no ideal-gas <pressure> dependence.
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