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I was taught that the the pre-exponential factor $A$ in the Arrhenius equation \eqref{arrhenius} does not increase with temperature.
$$k = A\cdot\mathrm \exp\left(-\frac{E_\mathrm a}{RT}\right)\tag1\label{arrhenius}$$
But, a component of the pre-exponential factor is the number of collisions, which presumably increases with temperature. Should the pre-exponential factor therefore not increase with temperature as well?
Your intuition is correct. The factor $A$ changes with temperature.
This article details how the value of $\ce{A}$ for an elementary, bimolecular reaction between $\ce{P}$ and $\ce{Q}$ can be derived to be:
$$A_{\ce{PQ}}=N_\ce{P}N_\ce{Q}d^2_{\ce{PQ}}\sqrt{\frac{8k_\mathrm{B}T}{\mu}}$$
The RHS is clearly a function of temperature. Without going into the details,[a] it suffices to remember that $A$ is a function of temperature because it is related to molecular collisions, which themselves are a function of temperature.
However, it is worth noting this paragraph on Wikipedia:
...under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the $\mathrm{e}^{(-E_\mathrm{a}/RT)}$ factor[b] (my emphasis); except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.
Given this, it may be within sufficient experimental errors to make the assumption that $A$ does not vary with temperature. However, it is just that, an assumption. In reality, $A$ does vary with temperature.
[a]: This isn't the exactly correct expression though. As the article itself notes, "Often times however, when the term is determined experimentally, $A$ is the preferred variable and when the constant is determined mathematically, $Z$ is the variable more often used. The derivation for $Z$, while mostly accurate, ignores the steric effect of molecules."
[b]: Of course, here Wikipedia is using the actually computed values of $A$ (and not vague estimated formulae like the one above; as actually $\sqrt{T}$ grows faster than $\mathrm{e}^{-1/T}$)
The Arrhenius equation for the rate constant is an empirical relationship and used only because it fits many sets of data. I'm assuming that you mean a temperature dependence in the prefactor $A$ as $T$ is already in the exponential.
Collision theory in the gas phase predicts an equation of the form $$k = a T^{1/2}\exp\left(\frac{-E_\mathrm a}{RT}\right),$$ where $a$ and $E_\mathrm{a}$ are fitted to the data. (If data is measured over a narrow range of temperatures (as is often the case) the $T^{1/2}$ has a small effect and $aT^{1/2}$ may appear to be constant with temperature and hence the 'normal' Arrhenius equation.) The $T$ in the exponential causes most of the change with temperature.
A more sophisticated derivation can be done using partition functions, which are defined in statistical mechanics, in which case the equation has the more general form $$k = aT^{\beta}\exp\left(\frac{-E_\mathrm a}{RT}\right),$$ where $a$ and $\beta$ depend on the particular reaction type (e.g. atom molecule, or molecule-molecule), $\beta$ can have positive and negative values usually in the range $-3/2$ to $2$. These values depend on knowing some properties of the transition state, which is the geometry of the species at the top of the barrier separating reactants to products. However, experimentally, these geometries are generally unknown. Thus there are three parameters to fit the data, $a$, $\beta$ and $E_\mathrm{a}$.
Just to mention in passing that, in solution, some reactions are 'diffusion controlled' reactions, whose rate constants are controlled by the viscosity of the solvent, not the activation energy which in these reactions is very small. A detailed theory due to Kramers describes the dragging ('friction') effect of a solvent on the reacting molecules. In the gas phase so called 'unimolecular reactions' are modelled by what is called RRKM theory (Rice-Ramspburger-kassel-Marcus) based also on statistical methods that describe how much energy the reactants have when they collide.
