  In the half-filled one-orbital Hubbard model on a square lattice, we study
the effect of next-nearest neighbor hopping on the single-particle spectral
function at finite temperature using an exact-diagonalization + Monte-Carlo
based approach to the simulation process. We find that the pseudogap-like dip,
existing in the density of states in between the N\'{e}el temperature $T_N$ and
a relatively higher temperature $T^*$, is accompanied with a significant
asymmetry in the hole- and particle-excitation energy along the high-symmetry
directions as well as along the normal-state Fermi surface. On moving from
($\pi/2, \pi/2$) toward $(\pi, 0)$ along the normal state Fermi surface, the
hole-excitation energy increases, a behavior remarkably similar to what is
observed in the $d$-wave state and pseudogap phase of high-$T_c$ cuprates,
whereas the particle-excitation energy decreases. The quasiparticle peak height
is the largest near ($\pi/2, \pi/2$) whereas it is the smallest near $(\pi,
0)$. These spectral features survive beyond $T_N$. The temperature window $T_N
\lesssim T \lesssim T^*$ shrinks with an increase in the next-nearest neighbor
hopping, which indicates that the next-nearest neighbor hopping may not be
supportive to the pseudogap-like features.
