  The physical mechanism behind the TeV gamma-ray source observed at the centre
of the Galaxy is still unknown. One intriguing possibility is that the
accretion flow onto the central supermassive black hole is responsible for
accelerating protons to TeV energies which then diffuse outward to interact
with molecular gas at distances of ~1 pc. Here, we build on our earlier
detailed calculations of the proton transport to consider the time and energy
dependence of the TeV signal following a burst of particle acceleration at Sgr
A*. We find that, due to the strong energy dependence of the proton diffusion,
any variability in the particle acceleration rate will only be visible in the
TeV signal after a delay of ~ 10 yrs, and only at energies >~ 10 TeV. If the
accelerator is long-lived it must have been running for at least 10^4 yrs and
have a hard proton injection spectrum of \alpha=0.7 (where dn/dE_inj \propto
E^{-\alpha}) in order to produce the correct amount of high energy gamma-ray
flux. This rapid diffusion of high energy protons also rules out the
possibility that the observed TeV source is directly related to the period of
increased activity of Sgr A* that ended ~ 100 yrs ago. However, a good fit to
the observed H.E.S.S. data was found with \alpha=2.7 for the scenario of a
brief (~few year long) burst of particle acceleration that occurred ~ 10 yrs
ago. If such bursts are common then they will keep the TeV source energised and
will likely produce spectral variability at >~ 10 TeV on <~ 5 yr timescales.
This model also implies that particle acceleration may be an important
mechanism in reducing the radiative efficiency of weakly accreting black holes.
