In a MPH model with gamma distributed unobserved heterogeneity, the hazard rates among survivors up to any given point in time are gamma distributed as well, but with smaller scale parameter. In a paper with Gerard van den Berg, I show that, much more generally, the distribution of the unobserved heterogeneity, appropriately rescaled, converges to a gamma distribution.  

Abbring, Jaap H., and Gerard J. van den Berg (2007), "The unobserved heterogeneity distribution in duration analysis", Biometrika, 94, 87–99.

In a large class of hazard models with proportional unobserved heterogeneity, the distribution of the heterogeneity among survivors converges to a gamma distribution. This convergence is often rapid. We derive this result as a general result for exponential mixtures and explore its implications for the specification and empirical analysis of univariate and multivariate duration models.

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