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The answer is yes under additional conditions on norm and conorm of $A$ and $\alpha$.

So, if

1. Periodic data conditions are satisfied. That is, for any periodic point $p$ $$ \sum_{x\in O(p)}f(x)=A\big(\sum_{x\in O(p)}g(x)\big)-\sum_{x\in O(p)}g(x). $$
2. Exponent $\alpha$ is sufficiently close to 1.
3. Transformation $A$ is dominated by $T$. That is, the map $(x,v)\mapsto(Tx, Av)$ is partially hyperbolic.

Then Walkden's paper "Solutions to the twisted cocycle equation over hyperbolic systems" proves that there exist an $\alpha$-Holder solution $g$. The result is more general.