Single-molecule biophysical tools permit measurements of the mechanical response of individual biomolecules to external load, revealing details that are typically lost when studied by ensemble methods. Kramers theory of diffusive barrier crossing in one dimension has been used to derive analytical solutions for the observables in such experiments, in particular, for the force dependent lifetimes. We propose a minimalist model that captures the effects of multidimensionality of the free energy landscape on the kinetics of a single-molecule system under constant applied force. The model predicts a rich spectrum of scenarios for the response of the system to the applied force. Among the scenarios is the conventional decrease in the lifetime with the force, as well as a remarkable rollover in the lifetime with a seemingly counterintuitive increase of the lifetime at low force followed by a decrease in the lifetime at higher forces. Realizations of each of the predicted scenario are discussed in various biological contexts. Our model demonstrates that the rollover in the lifetime does not necessarily imply a discrete switch between two coexisting pathways on the free energy landscape, and that the rollover can also be realized for a dynamics as simple as that on a single pathway with a single bound state. Our model leads to an analytical solution that reproduces the entire spectrum of scenarios, including the rollover, in the force-dependent lifetime, in terms of the microscopic parameters of the system.