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$$\omega(E) \sim \frac{ e^{ \beta E} }{ E^{ 1+D/2 } }$$

Is it possible to generalize this method in curved space?

A possible way is to calculate the torus path integral of a string that wraps the euclidian periodic time in a curved background. At high temperatures this can be calculated from the path integral of a single non-relativistic particle, which gives the free energy and thus the density of states. This seems to be called the random walk model. References: http://arxiv.org/abs/1506.07798 and http://arxiv.org/abs/hep-th/0508148 .

But this seems totally different. A particle path can be related to a random walk, but one doesn't calculate the number of microstates from combinatoric reasoning. Is there a way to do something like that?