Kepler's Second Law: conservation of the energy or of the angular momentum?

Multiply both sides of your equation by the mass and you have equality of the magnitude of the angular momentum at the perigee and the apogee. At these points the velocity is perpendicular to the position, so $|mathbf{L}|=|mathbf{r}times mmathbf{v}|=mvr$.

In modern language, Kepler’s Second Law expresses the conservation of angular momentum, not the conservation of energy. But it’s often stated in terms of geometry, as in Wikipedia:

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

To see the connection between this area-related statement of the Second Law and the conservation of angular momentum, recall that the magnitude of the cross product between two vectors is the area of the parallelogram that they form. One half of this is the area of the triangle that they form.

Imagine the triangle swept out over time $dt$ by the position vector $mathbf{r}$ as it changes by $mathbf{v}dt$. The infinitesimal area swept out is

$$dA=frac12|mathbf{r}timesmathbf{v}dt|$$

because one side of the triangle is $mathbf{r}$ and another side is $mathbf{v}dt$.

Thus

$$frac{dA}{dt}= |mathbf{r}timesmathbf{v}|=frac{|mathbf{L}|}{m}=text{constant}.$$

So the rate at which the area is swept out is constant because the angular momentum is constant.