  We consider the double pulsar binary system as a laboratory to locally test
the orbital effects induced by an uniform cosmological constant $\Lambda$ in
the framework of the known general relativistic laws of gravity, and the DGP
braneworld model of gravity independently of the cosmological acceleration
itself for which they were introduced. We, first, construct the ratio
R=Delta\dot\omega/Delta P of the discrepancies between the phenomenologically
determined periastron rate \dot\omega and orbital period and their predicted
values from the 1PN approximation and the third Kepler law. Then, we compare
its value R = (0.3 +/- 4) \times 10^-11 s^-2, compatible with zero within the
errors, to the ratios R_Lambda and R_DGP of the effects induced on the apsidal
rate and the orbital period by Lambda and the DGP gravity; we find them neatly
incompatible with R being R_Lambda = (3.4 +/- 0.3) \times 10^-8 s^-2 and R_DGP
= (1.4 +/- 0.1) \times 10^-7 s^-2, respectively. Such a result, which for the
case of Lambda is valid also for any other Hooke-like exotic force proportional
to r, is in agreement with other negative local tests recently performed in the
Solar System with the ratios of the non-Newtonian/Einsteinian perihelion
precessions for several pairs of planets.
