Brian Arthur and Paul David also have argued that economics should have room for non ergodic processes. They discuss such processes under the rubric of "path dependence". "Path Dependence - A Foundational Concept for Historical Social Science" is an example of a 2006 statement of Paul David's.
One can see that others thought neoclassicalism contained an assumption of ergodicity:
"Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the 'ergodic hypothesis' by analogy to the use of this term in statistical mechanics. Remember that the classical economists were fatalists (a synonym for 'believers in equilibrium'!). Harriet Martineau, who made fairy tales out of economics (unlike modern economists who make economics out of fairy tales), believed that if the state redivided income each morning, by night the rich would again be sleeping in their comfortable beds and the poor under the bridges. (I think she thought this a cogent argument against equilitarian taxes.)
Now, Paul Samuelson, aged 20 a hundred years later, was not Harriet Martineau or even David Ricardo; but as an equilibrium theorist he naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theorists hoped not to introduce hystersis phenomena into our model, as the Bible does when it says, 'We pass this way only once' and, in so saying, takes the subject out of the realm of science into the realm of genuine history." -- Paul A. Samuelson, "What Classical and Neo-Classical Monetary Theory Really Was," Canadian Journal of Economics, V 1., # 1, pp. 1-15, 1968. (Reprinted in Monetary Theory: Selected Readings, edited by Robert W. Clower, Penguin, 1969.)