  We investigate the origin of the difference, which was noticed by Fey {\it et
al.} [Phys. Rev. Lett. {\bf 104}, 145703 (2010)], between the steady state
density of an Abelian sandpile model (ASM) and the transition point of its
corresponding deterministic fixed-energy sandpile model (DFES). Being
deterministic, the configuration space of a DFES can be divided into two
disjoint classes such that every configuration in one class should evolve into
one of absorbing states, whereas no configurations in the other class can reach
an absorbing state. Since the two classes are separated in terms of toppling
dynamics, the system can be made to exhibit an absorbing phase transition (APT)
at various points that depend on the initial probability distribution of the
configurations. Furthermore, we show that in general the transition point also
depends on whether an infinite-size limit is taken before or after the
infinite-time limit. To demonstrate, we numerically study the two-dimensional
DFES with Bak-Tang-Wiesenfeld toppling rule (BTW-FES). We confirm that there
are indeed many thresholds. Nonetheless, the critical phenomena at various
transition points are found to be universal. We furthermore discuss a
microscopic absorbing phase transition, or a so-called spreading dynamics, of
the BTW-FES, to find that the phase transition in this setting is related to
the dynamical isotropic percolation process rather than self-organized
criticality. In particular, we argue that choosing recurrent configurations of
the corresponding ASM as an initial configuration does not allow for a
nontrivial APT in the DFES.
