Is Glorot/He-style variance-preserving *regularization* a known thing? In the context of deep learning, layer weight initialization techniques of the Glorot/He family strive to select initial random weight values in such a way that the variance of each layer's output is equal to the variance of its input. This helps avoid vanishing/exploding gradient issues and improves training stability.
It seems likely to me that this is a desirable property to maintain not only initially, but also during training, as a regularization constraint via an explicit loss term nudging weight distribution to the optimal distribution mandated by Glorot/He.
Has this been studied?
 A: 
Has this been studied?

Yes, it has been studied.  You are describing one of the goals of batchnorm.  Other methods exist as well - see @Sycorax's answer for another.
So in fact, it has been implemented in every major library, and used very extensively in research models and in deployed models.

variance of each layer's output is equal to the variance of its input

Maintaining activation variances over the course of gradient descent is one of the two mechanisms behind Batch Normalization (a.k.a. batchnorm.)  The other mechanism is to maintain activation means.  With batchnorm, each batch of training data, a translation and scale parameter is learned for each layer, such that the activations at each layer are re-centered and re-scaleed to track zero mean and unit variance.

as a regularization constraint.

Indeed, batchnorm has a regularizing effect.  The original goal of batchnorm was to reduce a phenomenon dubbed, "internal covariate shift," but more recent research suggests that it has a regularizing effect by smoothing the gradient steps, hence smoothing the effective loss function.
Figure 1 (c) of the latter paper shows layer 3 and layer 11 VGG activation distributions over the course of training, with and without batchnorm:

With batchnorm (blue), wee see that the distributions stay more consistent over training, especially in the tail behaviour, and especially in very early training.  In the bottom left, note how quickly the activation distributions narrow into tight variance and stay that way for the duration of training.
A: "Self-Normalizing Neural Networks" by Günter Klambauer Thomas Unterthiner Andreas Mayr & Sepp Hochreiter proposes a neural network that converges to activations with zero mean and unit variance.

Deep Learning has revolutionized vision via convolutional neural networks (CNNs)
and natural language processing via recurrent neural networks (RNNs). However,
success stories of Deep Learning with standard feed-forward neural networks
(FNNs) are rare. FNNs that perform well are typically shallow and, therefore cannot
exploit many levels of abstract representations. We introduce self-normalizing
neural networks (SNNs) to enable high-level abstract representations. While
batch normalization requires explicit normalization, neuron activations of SNNs
automatically converge towards zero mean and unit variance. The activation
function of SNNs are “scaled exponential linear units” (SELUs), which induce
self-normalizing properties. Using the Banach fixed-point theorem, we prove that
activations close to zero mean and unit variance that are propagated through many
network layers will converge towards zero mean and unit variance — even under
the presence of noise and perturbations. This convergence property of SNNs allows
to (1) train deep networks with many layers, (2) employ strong regularization
schemes, and (3) to make learning highly robust. Furthermore, for activations
not close to unit variance, we prove an upper and lower bound on the variance,
thus, vanishing and exploding gradients are impossible. We compared SNNs on
(a) 121 tasks from the UCI machine learning repository, on (b) drug discovery
benchmarks, and on (c) astronomy tasks with standard FNNs, and other machine
learning methods such as random forests and support vector machines. For FNNs
we considered (i) ReLU networks without normalization, (ii) batch normalization,
(iii) layer normalization, (iv) weight normalization, (v) highway networks, and (vi)
residual networks. SNNs significantly outperformed all competing FNN methods
at 121 UCI tasks, outperformed all competing methods at the Tox21 dataset, and
set a new record at an astronomy data set. The winning SNN architectures are often
very deep.

A: There are ways to preserve activation variance with an explicit regularization term.  (As opposed to implicit regularization via batchnorm, self-normalizing networks, etc.)  For example, the orthogonality regularizer
$$
\hat{\ell}(\theta) = \sum_k || {\theta_k}^T \theta_k - c_k I ||^2
$$
will do it, given square weight matrices $\theta_k$.  (Possibly for rectangular $\theta_k$ as well, but I can't recall or verify at the moment.)
Above, $\theta$ are network weights, with $\theta_k$ the weight matrix for each layer $k$.  $I$ is the identity matrix, and $c_k$ is a scalar gain factor depending on the chosen nonlinearity and on data distribution properties.  See details below.

Explanation
To state the goal a bit more formally, for layer $k$ in a neural net:

*

*Let $\theta_k$ be layer parameters, so that the input $x$ and output $y$ of the layer are written $y = f_{\theta_k}(x)$

*Let $X_k$ and $Y_k$ be random variables representing layer $k$ input and output, so that we can talk about their distributions.  i.e. $Y_k = f_{\theta_k}(X_k)$
Given a training sample $z$ and network weights $\theta = \{\theta_1, \theta_2, \dots\}$, we seek a regularization function $\hat{\ell}(\cdot)$ so that the overall objective function
$$
\mathcal{L}(z; \theta) = \underbrace{\ell(z; \theta)}_{\text{loss}} + \lambda \underbrace{\hat{\ell}(\theta)}_{\text{regularizer}}
$$
induces
$$
\text{Var}(Y_k) \approx \text{Var}(X_k)
$$
for each layer $k$, during and after optimization.

To start simply, assume that $\theta_k$ is a $d \times d$ square matrix and $f_{\theta_k}: \mathbb{R}^d \to \mathbb{R}^d$ is linear, so that the layer is just a matrix multiplication:
$$
\begin{align}
Y_k &= f_{\theta_k}(X_k) \\
&= \theta_kX_k
\end{align}
$$
In this case, enforcing orthogonality of $\theta_k$ is a simple way to preserve variance, regardless of the properties of $X_k$.  Enforcing or encouraging orthogonality during gradient descent is a well-studied problem.  Since orthogonality can be defined as ${\theta_k}^T \theta_k = I$, a simple regularizer just minimizes the distance between both sides of the equation for each layer:
$$
\hat{\ell}(\theta) = \sum_k || {\theta_k}^T \theta_k - I ||^2
$$
Hence, each weight matrix $\theta_k$ is pushed towards orthogonality, with the proximity to orthogonality controlled by regularization weight $\lambda$ in the objective function $\mathcal{L} = \ell + \lambda \hat{\ell}$
Note, this preserves variance for linear $\mathbb{R}^d \to \mathbb{R}^d$ layers only.  As we reintroduce practicality to our layers (i.e. nonlinear activation functions, non-square weight matrices, or even convolutions) then more care is necessary to preserve variance.  For example, if we assume $X_k \sim N(0, \sigma^2)$ and $f_{\theta_k}$ uses ReLU activation, then $\text{Var}(Y_k) = c \sigma^2$ for some constant $c$.  So we should replace the identity matrix $I$ with $\frac{1}{\sqrt{c}} I$ in the regularizer $\hat{\ell}(\cdot)$.  (I am too lazy to derive this particular $c$ right now, but empirically, $c \approx 0.34$.)  .
Different nonlinearities will have different gain factors.  For example, I think OPLU conveniently has a gain factor of 1.  These gain factors are discussed in a few different places, including the orthogonal initialization and descent literature, and the initialization literature in general.
Instead of soft-constraint by regularization, we can also hard-constrain to orthogonality by various methods.  For instance, this paper uses weight orthogonality to address vanishing/exploding gradients in recurrent networks.  But since you asked only about regularizers, I won't detail anything about that.  And typically, if one is interested in orthogonality-preserving optimization, then one is also interested in orthogonal weight initialization.
(The papers linked here are not comprehensive, and probably a bit out of date.  If anyone knows of relevant surveys to link instead, do share in the comments.)
