  We re-examine the black hole solutions in classical theories of dilaton
gravity in two dimensions. We consider an arbitrary dilaton potential such that
there are black hole solutions asymptotic at infinity to the nearly
$\mathrm{AdS}_2$ solutions of JT gravity, and such that the black hole energy
and entropy are bounded below. We show that if there is a black hole solution
with negative specific heat at some temperature $T$, then at the same
temperature there is a black hole solution with lower free energy and positive
specific heat. As the temperature is increased from 0 to infinity, the black
hole energy and entropy increase monotonically but not necessarily
continuously; there can be first order phase transitions, similar to the
Hawking-Page transition. These theories can also have solutions corresponding
to closed universes.
