  We perform Monte Carlo simulations of large two-dimensional Gaussian Ising
spin glasses down to very low temperatures $\beta=1/T=50$. Equilibration is
ensured by using a cluster algorithm including Monte Carlo moves consisting of
flipping fundamental excitations. We study the thermodynamic behavior using the
Binder cumulant, the spin-glass susceptibility, the distribution of overlaps,
the overlap with the ground state and the specific heat. We confirm that
$T_c=0$. All results are compatible with an algebraic divergence of the
correlation length with an exponent $\nu$. We find $-1/\nu=-0.295(30)$, which
is compatible with the value for the domain-wall and droplet exponent
$\theta\approx-0.29$ found previously in ground-state studies. Hence the
thermodynamic behavior of this model seems to be governed by one single
exponent.
