Let $T_1 \colon X_1 \to X_1$ be an Anosov diffeomorphism and let $T_2 \colon X_2 \to X_2$ be a uniquely ergodic expansive homeomorphism which is not a periodic orbit. Let $T \colon X_1 \times X_2 \to X_1 \times X_2$ be given by the direct product of the two maps. Clearly $T$ is expansive, and $T$ does not have specification because it has no periodic orbits. The invariant measures of $T$ are precisely the products of the invariant measures of $T_1$ with the unique invariant measure of $T_2$ (right...?). So, calculating the equilibrium state(s) of a Hölder function defined on $X$ is the same problem as calculating the equilibrium state(s) of the function on $X_1$ defined by integrating $f$ along fibers against the unique invariant measure of $T_2$. The fiberwise integral has to be Hölder because $f$ is Hölder, and it follows that $f$ has a unique equilibrium state.

Does that work?

Let $T_1 \colon X_1 \to X_1$ be an Anosov diffeomorphism and let $T_2 \colon X_2 \to X_2$ be a uniquely ergodic expansive homeomorphism which is not a periodic orbit. Let $T \colon X_1 \times X_2 \to X_1 \times X_2$ be given by the direct product of the two maps. Clearly $T$ is expansive, and $T$ does not have specification because it has no periodic orbits. The invariant measures of $T$ are precisely the products of the invariant measures of $T_1$ with the unique invariant measure of $T_2$ (right...?). So, calculating the equilibrium state(s) of a Hölder function defined on $X$ is the same problem as calculating the equilibrium state(s) of the function on $X_1$ defined by integrating $f$ along fibers against the unique invariant measure of $T_2$. The fiberwise integral has to be Hölder because $f$ is Hölder, and it follows that $f$ has a unique equilibrium state.

Edit: the sentence beginning "The invariant measures of $T$ are precisely..." is probably wrong - see comments below.