At a very fundamental level data storage in any device will decrease entropy locally (or bring in local order). Similar maybe the case with computing.
The above gets more difficult as the temperature rises because you need a sink to discharge the wate heat. So is there a thermodynamic limit after which computing/data storage may not work. Say the melting point of aluminum or iron or the temperature of the sun or a neutron star…
Yes, a black hole computer is the ultimate limit of computational density.
This Wiki page on “Landauer’s principle” (or this page) addresses OP’s question. This pdf studies the question as it applies to quantum computers.
The human kidney performs a computation task: it separates (or sorts) blood components into useful substances to be retained and waste to be excreted. It is said that the kidney is remarkably efficient, consuming energy near the thermodynamic minimum for the “computation” it performs.
I hope a mathematician or physicist comes along to summarize these ideas. 