  The Bekenstein-Hawking entropy of a black hole is proportional to its horizon
area, hence in $D=2$ spacetime dimensions it is constant because the horizon
degenerates into two points. This fact is consistent with Einstein's gravity
becoming topological in two dimensions. In $F(R)$ gravity, which is non-trivial
even in $D=2$, we find that the entropy is constant, as for Bekenstein-Hawking.
As shown in EPL 139 (2022) no.6, 69001 (arXiv:2208.10146), two-dimensional
$F(R)$ gravity is equivalent to Jackiw-Teitelboim gravity, in turn equivalent
to the Sachdev-Ye-Kitaev model where the entropy becomes constant in the large
$N$ limit. Several recently proposed entropies are functions of the
Bekenstein-Hawking entropy and become constant in $D=2$, but in two-dimensional
dilaton gravity entropies are not always constant. We study general dilaton
gravity and obtain arbitrary static black hole solutions for which the
non-constant entropies depend on the mass, horizon radius, or Hawking
temperature, and constitute new proposals for a generalized entropy.
