Although I've forgotten the proof (and cannot find it in, say, Carrol's book), the following formula holds for the covariant divergence in general relativity:

$$\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt{|g|}} \partial_{\mu} \left( \sqrt{|g|} A^{\mu}\right).$$

where $g = \det(g_{\alpha\beta})$. I was wondering if this formula holds if $A^{\mu}$ is replaced with a general rank $(n,m)$ tensor

$$T^{\mu \mu_1\mu_2 \cdots \mu_{n-1}}_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nu_1\cdots \nu_m}?$$

If not, could you point me to any references that have divergence formulas for higher rank tensors?

Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:

$$\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt{|g|}} \partial_{\mu} \left( \sqrt{|g|} A^{\mu}\right),$$

where $g = \det(g_{\alpha\beta})$. I was wondering if this formula holds if $A^{\mu}$ is replaced with a general rank $(n,m)$ tensor

$$T^{\mu \mu_1\mu_2 \cdots \mu_{n-1}}_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nu_1\cdots \nu_m}?$$

If not, could you point me to any references that have divergence formulas for higher rank tensors?