Parallelizing Polyhedral volume computation

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For an application, I need to compute volumes of convex polytopes of dimension of about 300. The polytopes of interest have a restricted form:

$$P:={ (x_1,x_2,\ldots,x_n) } ~\text{with} \ 0\leq x_i\leq x_{i+1} \leq L,\ l\leq x_i - x_{i+1} \leq u$$ where $L,u,l$ are known positive constants.

On my laptop, the polytope volume computation routine takes about 3 hours to compute for $n=9$, and freezes for larger $n$, making it useless for most interesting cases of my application.

Is there an easy way to parallelize the volume routine?

Perhaps by a straightforward use of the [multiprocessing] module, when running on a cluster ?

A more advanced use case, which also arises in my application, lies in computing integrals over the polytope: $\int_{P} f(\mathbf{x}) d\mathbf{x}$, where $f$ is a symmetric function of its arguments. I would greatly be interested in techniques that extend to this case as well.