If the Sun were bigger but colder, Earth would be hotter or colder? That is the question. I know the concepts of luminosity $(L=4\pi R^2F)$ and the flow $F=\sigma· T^4$, with $T$ the temperature in its surface. But how I use that to know if the Earth would get warmer or not if $R$ increases and $T$ decreases?
The equilibrium temperature of the Earth, $T_E$, scales roughly as $L^{1/4}$, which is proportional to $R^{1/2} T$, where $L$, $R$ and $T$ are the solar luminosity, radius and temperature.

The actual approximate relationship is derived by equating the power received by the Earth, which is proportional to the solar luminosity $L$, with the power radiated by the Earth, which is proportional to $T_E^4$ for a blackbody. Hence $T_E \propto L^{1/4}$.

So the answer to your question depends on by how much you increase the radius compared with the decrease in temperature.

There will be second order effects that do depend on the spectrum of radiation from the Sun (and therefore its temperature) compared with the wavelength dependence of the albedo and emissivity of the Earth. So I will post a better question...