Luminosity in polar regions I tried to find the habitable planets, thus I suppose to calculate the effective temperatures that incident on the planets from their host stars, I used 

$$
T_\mathrm{eff}=\frac{L}{4\pi\sigma_\mathrm{SB}R^2}
$$

but I don't know how to calculate the effective temperatures in polar regions?
The formula you have written resembles the Stefan-Boltzmann law, and its correct form is $$T^4 = \frac{L}{4\pi R^2 \sigma_{SB}}$$.  It relates the luminosity of a blackbody of a given radius to its effective temperature. Earth as a whole can be assumed to be a black body at an effective temperature. So you can calculate the effective temperature of the "earth". This is very well explained in the example section of this wiki page: https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

For calculating the effective temperature of the polar region, first assumption is that polar region is a black body. Then the question to ask is: what is the effective teperature of the polar region given that the it reaches equilibrium when the Sun light falls on it. To calculate this further, you would need to calculate the angle at which the sun light fall on the polar region you are interested in and also the distance travelled by the sun light in the atmosphere. These parameters will vary with respect to the time of the year, time of the day and the region you are interested in (North pole, South pole). Some calculated temperatures are given on the same wiki page. I hope that helps.