  The thermodynamics of charged topological black holes (TBHs) with different
horizon geometries in $d$-dimensional Einstein-Maxwell and 4-dimensional
conformal gravities is revisited using the restricted phase space formalism.
The concept of subsystems for black holes is introduced, which enables a
precise description for the thermodynamic behaviors of (non-compact) black
holes with infinitely large horizon area. The concrete behaviors can be
different for TBHs in the same underlying theory but with different horizon
geometries, or for those with the same horizon geometry but from different
underlying theories.
  In some concrete thermodynamic processes, the non-compact black holes in
4-dimensional conformal gravity can reach certain state with zero entropy but
nonvanishing temperature, which is physically unsounded. This suggests a novel
use of black hole thermodynamics as a tool for testing the viability of gravity
models or constraining the model parameters or integration constants. The high
and low temperature limits of the TBHs are also considered, and it is found
that, at high temperature, all TBHs behave like low temperature phonon gases,
while at low temperature, all charged as well as neutral hyperbolic TBHs in
Einstein-Maxwell gravity behave like strongly degenerated electron gases.
Finally, using the concept of subsystems, some conceptual issues in the
description for thermal fluctuations in black hole systems are clarified, and
the relative thermal fluctuations for finite subsystems are also analyzed in
some detail.
