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

Static pure Lovelock black hole solutions with horizon topology ${\bf S^{(n)} \times S^{(n)}}$