  We study the intrinsic Hall conductivity of the ordinary and topological
superconducting phases of a Rashba metal in a perpendicular Zeeman field. In
this system the normal metal breaks time reversal symmetry while the
superconducting order parameter does not, in contrast to the chiral p-wave
superconducting state predicted in the monolayer strontium ruthenate
(Sr$_2$RuO$_4$) whose Hall conductivity has been studied extensively. We study
the effects of intra-band and inter-band pairing and find there is
qualitatively larger change in the intrinsic Hall conductivity when there is
inter-band pairing, with the change in magnitude linear in the pairing gap. We
argue that inter-band pairing leads in general to higher energy costs for the
topological phase compared to the topologically trivial phase and thus that the
qualitative behavior of the intrinsic Hall conductivity with superconductivity
in these systems could provide important clues about the nature of pairing in
the superconducting phase and even some hints of whether it is topological or
not.
