Is the variance of a trait related to its rate of evolution?
The question can be understood in different ways
Causality
I think the question is a bit unclear in using the term associated in terms of what is being causal. Is the interest to understand how do processes that affect speed of evolution also affect phenotypic variance or is the question `how does a change in phenotypic variance affect the speed of evolution.
What is meant by speed of evolution
Speed of evolution has no strict definition. Or rather it has two standard definitions and it can be measured in Haldanes or in Darwins (which are units).
Quite often by &quot;speed of evolution&quot; authors mean &quot;response to selection over medium to long term&quot; as for example in the wiki article for rate of evolution.
Below I am answering to two different questions.
How does processes that affect &quot;speed of evolution&quot; at neutral sequences also affect phenotypic variance
At neutral sequences, the rate at which mutations fixes is equal to the mutation rate (see this answer for explanation). And all else being equal (typically without changing the mutational effect), high mutation rate leads to high phenotypic variance.
How does a sudden change in the phenotypic variance affect the response to selection in regard to this trait?
I think you pretty much said it all. I will just rephrase what you just said with a few simple equations.
Breeder's equation and heritability
The response $R$ to selection $S$ is given by
$$R=S h_N^2$$
, where $h_N^2$ is the heritability in the narrow sense. This above equation is known as the breeder's equation and is a fundamental result from quantitative genetics. The heritability in the narrow sense is defined as
$$h_N^2 = \frac{V_A}{V_P}$$
, where $V_A$ is the additive genetic variance and $V_P$ is the phenotypic variance.
Partitioning phenotypic variance
So it feels like increasing $V_P$ would decrease $h_N^2$ and therefore the response to selection $R$, but that would be very naive as $V_P$ is not independent of $V_A$. It is quite standard to decompose the phenotypic variance into genetic ($V_G$) and environment variances ($V_E$) $\left(V_P = V_G + V_E\right)$. Of course in reality, we should take into account variance due to other factors such as epistasis or developmental noise for example and all of their covariances. The genetic variance can further be decomposed into additive genetic variance $V_A$ and dominance genetic variance $V_D$ $\left(V_G = V_A + V_D\right)$.
Interdependence between $V_P$ and $V_A$
$V_P$ is a function of $V_A$. So as currently stated, it is impossible to answer the question by a simple yes or no. The shortest correct answer would probably be &quot;The response to selection will decrease if the increased variance is not (not mainly) related to increase in additive genetic variance&quot;.
To be perfectly accurate, let $\Delta_P$ be the increase phenotypic variance and $\Delta_A$ be the increased genetic additive variance such that $\Delta_P &gt; \Delta_A$. All else being equal the increase in $V_P$ will cause faster response to selection if and only if $\frac{Va+\Delta_A}{V_P+\Delta_P} &gt; \frac{Va}{V_P}$. Isolating the term of interest $\Delta_P$, we got
$$\Delta_P &lt; \frac{\Delta_A V_P}{V_A} = \frac{\Delta_A}{h_N^2} $$
In other words, all else being equal, an increase of phenotypic variance of $\Delta_P$ above the original variance $V_P$ increases the response to selection if and only if this increase is lower than the increase in additive genetic variance $\Delta_A$ multiplied by the inverse of the original heritability (heritability before a change in phenotypic variance occured).&quot;