What's the effect of temperature on buffering capacity?

Mathematically

For a buffer solution containing a weak acid and its salt with a strong base, buffer capacity is given by $$\beta = 2.303 ( [\ce{H+}] + [\ce{OH-}] + \frac{C_{buff}.K_{\mathrm{a}}.[\ce{H+}]}{([H^+]+K_{\mathrm{a}})^2})$$ It should be noted that the third term in the expression, $$\frac{C_{buff}.K_{\mathrm{a}}.[\ce{H+}]}{([\ce{H+}]+K_{\mathrm{a}})^2} = \frac{[\ce{HA}][{\ce{A^-}}]}{[\ce{HA}]+[\ce{A-}]}$$ which is independent of temperature ( if you are working at same concentrations at any temperature ) and the only temperature dependent terms are $\ce{[H+]}$ and $\ce{[OH-]}$ and $${[\ce{H+}] = \frac{K_{\mathrm{a}}.C_{acid}}{C_{salt}}}$$ and $${[\ce{OH-}]= \frac{K_w.C_{salt}}{K_a.C_{acid}}}$$ For the sake of simplicity let us consider the maximum buffer capacity at a given temperature, which occurs when $${C_{salt} = C_{acid}}$$ Now, $${\beta= 2.303( K_a + \frac{K_w}{K_a} + \frac{C_{buff}}{4})}$$ The acid dissociation constant $K_{\mathrm{a}}$ varies with temperature approximately as ${e^{\frac{-\Delta H_{ion,acid}}{RT}}}$ and $K_{\mathrm{w}}$ as ${e^{\frac{-\Delta H_{self-ion,water}}{RT}}}$.

Differentiating $${\beta= 2.303( K_a + \frac{K_w}{K_a} + \frac{C_{buff}}{4})}$$ with temperature, and substituting values of ${\Delta H_{ionization,acid}}$ and ${\Delta H_{self-ionization,water}}$, for most of the acids, we get a graph like this: It increases up to a certain temperature, depending on the acid and then it starts to decrease.

Experimental Observations

For Histidine and related amino acids the variation is shown below:

For further details read, Effect of Temperature on the Buffering Capacities of Histidine-Related Compounds and Fish Skeletal Muscle^[1].

Reference:

[1]: Abe, H.; Okuma, E. Effect of Temperature on the Buffering Capacities of Histidine-Related Compounds and Fish Skeletal Muscle. Nippon Suisan Gakkai Shi 1991, 57 (11), 2101–2107.

You question is too broad to answer since there is no "universal answer." In chemistry there is a "tends too" answer and then a list of exceptions.

By and large the root of the question centers around the change in the equilibrium constant as a function of temperature. The equilibrium constants do depend on temperature, but how much "tends" to depend on the particular equilibrium under consideration. However the Arrhenius equation can offer some insight.

The other assumption is that temperature has no effect on buffering capacity since buffering capacity apparently only depends on two factors: concentration of the buffer solution and the ratio of pH to pKa. So even if the pKa changes due to the change in temperature, the pH also changes accordingly whereby pH increases when pkA increases and vice versa. So the ratio will remain the same no matter what.

This is the better argument. For example, if you start with equal concentrations of conjugate acid and base, and change the temperatures, that ratio will not change much, and the buffer capacity would be similar. The pH will be different as the pKa will change, but you already mentioned that.

There are small secondary effects: The volume of the solution might increase slightly, slightly decreasing the buffer concentration and its buffer capacity. Also, the change in pKa will shift the equilibrium a bit, resulting in the change of pH and in a slight change in the ratio of conjugate acid and base.