  An extremely compact object (ECO) is defined as a quantum object without
horizon, whose radius is just a small distance $s$ outside its Schwarzschild
radius. We show that any ECO of mass $M$ in $d+1$ dimensions with $s\ll
(M/m_p)^{2/(d-2)(d+1)}l_p$ must have (at leading order) the same thermodynamic
properties -- temperature, entropy and radiation rates -- as the corresponding
semiclassical black hole of mass $M$. An essential aspect of the argument
involves showing that the Tolman-Oppenheimer-Volkoff equation has no consistent
solution in the region just outside the ECO surface, unless this region is
filled with radiation at the (appropriately blueshifted) Hawking temperature.
In string theory it has been found that black hole microstates are fuzzballs --
objects with no horizon -- which are expected to have a radius that is only a
little larger than the horizon radius. Thus the arguments of this paper provide
a nice closure to the fuzzball paradigm: the absence of a horizon removes the
information paradox, and the thermodynamic properties of the semiclassical hole
are nonetheless recovered to an excellent approximation.
