Can the temperature fluctuate in a system described by the Boltzmann distribution?

From what I understand, for a system A in thermal equilibrium with a heat bath B at temperature $T$, the Boltzmann distribution gives the probability of finding $A$ in state $i$. Different states have different energies (which determines their probability), hence there will be energy fluctuations in A. My question is whether the fluctuations in $E_A$ give rise to corresponding temperature fluctuations in $T_A$.

On the one hand, by definition two systems in thermal equilibrium have the same temperature, so $T_A = T_B equiv T$. Therefore the temperature $T_A$ does not fluctuate. And in almost everything I’ve read, $T$ is treated as a constant in the Boltzmann distribution.

On the other hand, each energy level $E_A$ has associated with it a certain number of states of $A$ with energy $E_A$. Thus $E_A$ has a corresponding entropy $S_A = f(E_A)$, with $frac{dS_A}{dE_A} = f'(E_A) = 1/T_A$. Therefore, fluctuations in $E_A$ could give rise to fluctuations in $S_A$ and $T_A$.

How does one reconcile these? Does it have to do with the fact that in deriving the Boltzmann distribution one expands the heat bath entropy $S_B$ only to first order in $E_A$ by assuming $E_A$ is small enough that $Delta S_B approx frac{dS_B}{dE_A}Delta E_A$ and also $Delta S_A approx frac{dS_A}{dE_A}Delta E_A$? If that’s the case, does Boltzmann only hold for small fluctuations in $E_A$? Or is there an implicit average or another aspect of this formulation I am not properly accounting for?