  It is well known that vacuum equation of arbitrary Lovelock order for static
spacetime ultimately reduces to a single algebraic equation, we show that the
same continues to hold true for pure Lovelock gravity of arbitrary order $N$
for topology ${\bf S^{(n)} \times S^{(n)}}$. We thus obtain pure Lovelock
static black hole solutions with two-sphere topology for any order $N$, and in
particular we study in full detail the third and fourth order Lovelock black
holes. It is remarkable that thermodynamical stability of black hole discerns
between odd and even $N$, and consequently between negative and positive
$\Lambda$ and it favours the former while rejecting the latter.
