When I look for cartesian closed monads, I only find monads where the endofunctor preserves the cartesian structure of a cartesian closed category

$$ \operatorname T\ (a \times b) = (\operatorname T\ a) \times (\operatorname T\ b). $$

Are there any monads that also preserve the closed structure of a cartesian closed category, so that

$$ \operatorname T\ (a^b) = (\operatorname T\ a) ^{\operatorname T\ b}, $$

where $a^b$ denotes the exponential object of $a$ and $b$ in the category?