This is the other way around: the thermodynamic arrow of time is sometimes invoked to explain why we have memories from the past and not from the future.
The thermodynamic arrow of time rests on the idea that past sates are statistically implausible (this is the past hypothesis) and that implausible states have more chance to evolve into more plausible states. Now consider a footstep in the sand: this is a rather implausible state. Chances are that this state evolved from an even more implausible state in the past (someone walking on the sand). So traces are always traces of a more implausible state, i.e. traces from the past. The same would go for our memories, construed as traces from the past.
It is controversial whether such explanation is circular or not. Some (e.g. Maudlin) argue that it already requires a notion of evolution, which will be directed from past to future, so the explanation already assumes an arrow of time. It is also controversial whether the applicability of the notion of entropy, which measures the implausibility of a state, can be extended to these cases which are not strictly speaking in the realm of thermodynamics. However some authors (e.g. Albert) argue that a past hypothesis is available in any case, even if it is not casted in terms of entropy.