  We propose a new theoretical approach for a cosmological model, which starts
from an exponential of the trace of the energy-momentum tensor-dependence on
the gravitational action, to be summed to the Ricci scalar. We derive the
referred field equations and Friedmann-like equations. We derive the scale
factor, Hubble parameter and deceleration parameter, in terms of both time and
redshift. In possession of those parameters in terms of the redshift, we
confront their predictions with the observational Hubble dataset and the
outcomes are pretty satisfactory so that the model can be seen as a new
alternative to the cosmological constant problem. We also present the
statefinder diagnostic and discuss profoundly the dynamical behaviour of the
model.
