  Our main goal in this paper is to answer new positive cases of the natural
generalized version of Hartshorne's celebrated question on cofiniteness of
local cohomology modules, and consequently of Huneke's conjecture on the
finiteness of their sets of associated primes. Our approach, by means of which
we extend several results from the literature, is essentially based on spectral
sequence techniques and connections to numerical invariants such as the
cohomological dimension and the Gorenstein projective dimension. We also
provide, over a polynomial ring, a rather pathological example of a non weakly
Laskerian module (i.e., it admits a quotient with infinitely many associated
primes) whose first local cohomology module is non-zero and has finitely many
associated primes.
