  Motivated by the recent resurgence of interest in topological
superconductivity, we study in this paper superconducting pairing instabilities
of the hole-doped Rashba-Hubbard model on the square lattice with first- and
second-neighbor hopping. Within a weak-coupling approach based on the random
phase approximation, we compute the spin-fluctuation mediated paring
interactions as a function of spin-orbit coupling and hole doping. Rashba
spin-orbit coupling splits the spin degeneracies of the bands, which leads to
two van Hove singularities at two different fillings. We find that for a broad
doping region in between these two van Hove fillings the spin fluctuations
exhibit a strong ferromagnetic contribution. Because of these ferromagnetic
fluctuations, the triplet $f$-wave pairing channel is dominant within this
filling region, resulting in a topologically nontrival phase. We discuss
possible experimental realizations of this phase in heavy-fermion hybrid
structures.
