  The thermodynamic properties of self-gravitating spherical thin matter shells
an black holes in $d>4$ dimensions are studied, extending previous analysis for
$d=4$. The shell joins a Minkowski interior to a Tangherlini exterior, i.e., a
Schwarzschild exterior in $d$ dimensions, with $d\geqslant4$, The junction
conditions alone together with the first law of thermodynamics enable one to
establish that the entropy of the thin shell depends only on its own
gravitational radius. Endowing the shell with a power-law temperature equation
of state allows to establish a precise form for the entropy and to perform a
thermodynamic stability analysis for the shell. An interesting case is when the
shell's temperature has the Hawking form, i.e., it is inversely proportional to
the shell's gravitational radius. It is shown in this case that the shell's
heat capacity is positive, and thus there is stability, for shells with radii
in-between their own gravitational radius and the photonic radius, i.e., the
radius of circular photon orbits, reproducing unexpectedly York's thermodynamic
stability criterion for a $d=4$ black hole in the canonical ensemble.
Additionally, the Euler equation for the matter shell is derived, the
Bekenstein and holographic entropy bounds are studied, and the large $d$ limit
is analyzed. Within this formalism the thermodynamic properties of black holes
can be studied too. Putting the shell at its own gravitational radius, i.e., in
the black hole situation, obliges one to choose precisely the Hawking
temperature for the shell which in turn yields the Bekenstein-Hawking entropy.
The stability analysis implies that the black hole is thermodynamically stable
substantiating that in this configuration our system and York's canonical
ensemble black hole are indeed the same system. Also relevant is the derivation
in a surprising way of the Smarr formula for black holes in $d$ dimensions.
