  We bring forward a unified framework for the study of the superfluid
stiffness and the quantum capacitance of superconducting platforms exhibiting
conventional spin-singlet pairing. We focus on systems which in their normal
phase contain topological band touching points or crossings, while in their
superconducting regime feature a fully gapped energy spectrum. Our unified
description relies on viewing these two types of physical quantities as the
charge current and density response coefficients obtained for ``slow"
spatiotemporal variations of the superconducting phase. Within our adiabatic
formalism, the two coefficients are given in terms of Berry curvatures defined
in synthetic spaces. Our work lays the foundations for the systematic
description of topological diagonal superfluid responses induced by
singularities dictating the synthetic Berry curvatures. We exemplify our
approach for concrete one- and two-dimensional models of superconducting
topological (semi)metals. We discuss topological phenomena which arise in the
superfluid stiffness of bulk systems and the quantum capacitance of Josephson
junctions. We show that both coefficients become proportional to a topological
invariant which counts the number of topological touchings/crossings of the
normal phase band structure. These topological effects can be equivalently
viewed as manifestations of chiral anomaly. Our predictions are experimentally
testable in graphene-superconductor hybrids.
