  We show universality of isothermal fluid spheres in pure Lovelock gravity
where the equation of motion has only one $N$th order term coming from the
corresponding Lovelock polynomial action of degree $N$. Isothermality is
characterized by the equation of state, $p = \alpha \rho$ and the property,
$\rho \sim 1/r^{2N}$. Then the solution describing isothermal spheres, which
exist only for the pure Lovelock equation, is of the same form for the general
Lovelock degree $N$ in all dimenions $d \geq 2N+2$. We further prove that the
necessary and sufficient condition for the isothermal sphere is that its metric
is conformal to the massless global monopole or the solid angle deficit metric,
and this feature is also universal.
