There is considerable evidence of a positive correlation of standard of living and democracy across countries. Given this, it would seem natural that growth in the standard of living would be positively correlated with democratic change. One could even conjecture that there is a causation from growth in GDP to democratic change. Theory, though, indicates that this causation could be positive or negative: sluggish growth may lead to popular uprising and democratization, or booms may raise political expectations.
Paul Burke and Andrew Leigh explore this question with data from a large set of countries. Causation is not that obvious to measure well, because of the obvious endogeneity problem: democratization may also cause growth. They use some well-reasoned instruments to cover this problem, but I want to retain here only results with temperature as an instrument, as it is the only one that does not turn out to be statistically weak. And the causation is negative: You cannot expect that healthy growth can also lead to democratization. While there are always exceptions, do not keep your hopes of democratization in China too high.