What may be a bigger factor in Russian insecurity over the EaP may not be the set-up of the EaP itself, which is economically-orientated, but the weakness of Russia to maintain its sphere of influence. Less than a year ago it was harder to see the shrinkage of Russian influence with the war in the Caucasus, which asserted Russian military dominance in the region. However, it was interesting to see that Belarus didn't recognise the breakaway republics in line with Russian foreign policy.
Now it seems that Belarus is moving further from the Russian sphere due to a list of grievences with Russia:
"Mr Lukashenko has been frustrated by the Kremlin’s reticence over plans to form a “union state” linking Russia and Belarus and by energy disputes, but he was infuriated by Moscow’s refusal this week to provide finance to his country due to fears it is badly run and could collapse."
Russia's reluctance to cultivate its influence in its backyard - probably due to the pressure the economic crisis has put it under - could lead to Belarus becoming less able to play the EU and Russia off each other in the short term, as the Russian card looses some of its value. Hard power may be impressive, but its utility can often be very constrained, and Russian hard power isn't always that impressive.
That said, it remains to be seen if the EU can really fill the vaccum through the EaP, or if the political will and resources are there to play the influence game. Especially at a time when showing solidarity with the big bang EU members can be problematic, never mind helping out a country widely considered to be "the last dictatorship in Europe". It could turn out to be a temporary spat and Belarussan relations with Moscow could warm up again.