Thermodynamics entropy and temperature
In scenarios like the one you describe, the surroundings are typically modeled as an infinite reservoir, with the capacity to supply or absorb heat with negligible change in temperature.  If you treated it as a large finite reservoir instead, and evaluated its entropy change based on $\Delta S=mc\ln ((T_0+\Delta T)/T_0)$ and its heat absorbed by $Q=mc\Delta T$ (where m is the mass of the reservoir and c is its heat capacity), you would find that, in the mathematical limit of "mc" becoming infinite (at constant Q), the entropy change would approach $\Delta S=Q/T_0$.