The underlying event activity in proton-proton collisions at forward pseudorapidity (-6.6 < eta < -5.2) is studied with the CMS detector at the LHC, using a novel observable: the ratio of the forward energy density, dE/d(eta), for events with a charged-particle jet produced at central pseudorapidity (vertical bar eta(jet)vertical bar < 2) to the forward energy density for inclusive events. This forward energy density ratio is measured as a function of the central jet transverse momentum, p(T), at three different pp centre-of-mass energies (root s = 0.9, 2.76, and 7 TeV). In addition, the root s evolution of the forward energy density is studied in inclusive events and in events with a central jet. The results are compared to those of Monte Carlo event generators for pp collisions and are discussed in terms of the underlying event. Whereas the dependence of the forward energy density ratio on jet p(T), at each root s separately can be well reproduced by some models, all models fail to simultaneously describe the increase of the forward energy density with root s in both inclusive events and in events with a central jet.