  In this paper we present a Monte Carlo calculation of the critical
temperature and other thermodynamic quantities for the unitary Fermi gas with a
population imbalance (unequal number of fermions in the two spin components).
We describe an improved worm type algorithm that is less prone to
autocorrelations than the previously available methods and show how this
algorithm can be applied to simulate the unitary Fermi gas in presence of a
small imbalance. Our data indicates that the critical temperature remains
almost constant for small imbalances $h=\Delta\mu/\epsilon_F\lessapprox0.2$. We
obtain the continuum result $T_c=0.171(5)\epsilon_F$ in units of Fermi energy
and derive a lower bound on the deviation of the critical temperature from the
balanced limit, $T_c(h)-T_c(0)>-0.5\epsilon_Fh^2$. Using an additional
assumption a tighter lower bound can be obtained. We also calculate the energy
per particle and the chemical potential in the balanced and imbalanced cases.
