Simple CubeMap opengl wrapper class
It looks fine. Simple but gets the job done. A few things to look into:


GL_TEXTURE_CUBE_MAP_TYPES constant array could be moved to the source file, to hide that implementation detail, assuming it is not meaningful outside the class. I would also use a different name. It looks too much like an OpenGL constant, so it might be misleading. ALL_UPPERCASE is also usually associated with macro names. Not the case here.
This #include &lt;alpha/CubeMap.h&gt;, it should probably be between quotes, since it is a local header file of your project.
Since you are storing all those pos...File parameters of the constructor into std::strings, you might as well take an array of strings, or even a vector/array to be more C++-strict. Or at least replace the char*s by strings.
The correct signature of the assignment operator is:

CubeMap&amp; operator = (const CubeMap&amp; other) = delete;
^^^^^^^^ 


Returning a reference to the class type. Even though deleted, declare it properly to make sure the compiler recognises it as the default assignment op.
To avoid having to write the lengthy and error prone (sizeof(GL_TEXTURE_CUBE_MAP_TYPES) / sizeof(GL_TEXTURE_CUBE_MAP_TYPES[0])) you can use std::array (C++11), which has a size() method, or define a template function to make it less verbose:

template&lt;class T, std::size_t N&gt;
constexpr std::size_t arrayLength(const T (&amp;)[N])
{
    return N;
}

That assert(0) in the error case after stbi_load seems a bit haphazard. If the image fails to load on "release", you'll pass a null pointer to glTexImage2D. That will still allocate a texture image with undefined contents. That's not really helpful to the user. The error is already being logged, so you could either set a default placeholder image to visually signify the error, or simpler, just throw an exception and let the caller decide what to do.
That 31 is very hardcoded in assert(unit &gt;= 0 &amp;&amp; unit &lt;= 31). There's actually a maximum value you can query with glGetInteger, which I don't recall now, but it is quite large, so you can indeed be lazy and use a constant instead if you want. But it would still be better to define a MaxTextureUnits somewhere to make the value self-explanatory for the non-OpenGL versed.