Suppose I have a quadrature method with a theoretical error estimate that scales with the number of points as $e(N)$:
$$\int_a^b f(x)\mathrm{d}x = \sum_{i=1}^N w_i f(x_i) + \mathcal{O}\bigl(e(N)\bigr)$$
For example, $e(N) = N^{-4}$ for Simpson's rule, or $e(N) = N^{-2}$ for the trapezoidal rule. In this question I only care about how the error estimate scales with $N$, not its dependence on properties of the function or any leading coefficients. (And yes, I know it's just an estimate.)
If I use this quadrature method to perform a $d$-dimensional integral by repeated 1D integration,
$$\int_{a_0}^{b_0} \int_{a_1(x_0)}^{b_1(x_0)} \cdots \int_{a_d(x_0,\ldots,x_{d-1})}^{b_d(x_0,\ldots,x_{d-1})} f(\vec{x})\mathrm{d}^d\vec{x}$$
how does the error estimate for the multidimensional integral scale with $N$, the number of function evaluations along each dimension? Is it just $e(N)$? Or some function of $e(N)$? (like $[e(N)]^d$) Is it impossible to give a general relation without knowing the specific quadrature method in use?
I've done some Google searching and checking in Numerical Recipes but I can't find a straightforward answer to this. I would have thought it would be common.