This work is devoted to the numerical solution of the Ripa equations, a shallow-water-type model derived from the compressible Euler equations in order to incorporate horizontal temperature gradients. We propose a splitting strategy that decouples the acoustic and transport phenomena present in the system. Specifically, we first solve the pressure subsystem using an implicit scheme and then treat the transport subsystem explicitly. The use of semi-implicit schemes instead of fully explicit ones allows for less restrictive time-step constraints, especially in low-Froude-number regimes. We construct finite volume schemes that preserve all hydrostatic steady states, that is, steady states corresponding to zero velocity, and not only those associated with water at rest and/or isobaric equilibria. While in the shallow water case the steady states can be determined explicitly, this is not the case for the Ripa model. For this reason, we prescribe a profile for one of the variables and compute the remaining one accordingly using a collocation method. Finally, we present several numerical experiments to illustrate the good performance of the proposed approximations. For this purpose, we compare our method with two other splitting schemes: one that is not well-balanced, and another that is exactly well-balanced only for the water-at-rest and isobaric steady states, but not for general hydrostatic equilibria.