Say I have a Hadamard $d$-manifold $M$ with an upper Ricci curvature bound of $-b^2$. Write the volume form in polar exponential coordinates at $p\in M$ as $V(r,\theta) \, dr \, d\theta$, and similarly write the volume form of the simply connected $d$-manifold with constant curvature $-b^2$ as $V_{const}(r)\, dr\, d\theta$ in exponential coordinates.

Is it true that the ratio $V(r,\theta)/V_{const}(r)$ is bounded away from zero?

There's a true analogous statement for lower Ricci bounds, see e.g. Eq 4.2 of http://mail.math.ucsb.edu/~wei/paper/06survey.pdf.

140(1994) 655-683) that any (paracompact, Hausdorff, connected) smooth manifold of dimension $d\geq 3$ admits a Riemannian metric with negative Ricci curvature, scalar curvature bounded above by -1 and finite volume. On the other hand, lower bounds on Ricci curvature may have strong implications for the topology of the underlying manifold, which stem from Bishop-Gromov-type volume comparison estimates such as the one you stated. $\endgroup$ – Pedro Lauridsen Ribeiro May 17 '16 at 6:32