Summary: Let \(G\) be a simple graph with no isolated vertex and let \(\gamma_{tdR}(G)\) be the total double Roman domination number of \(G\). In this paper, we present lower and upper bounds on \(\gamma_{tdR}(G)\) of a graph \(G\) in terms of the order, open packing number and the numbers of support vertices and leaves, and we characterize all extremal graphs. We also prove that for any connected graph \(G\) of order \(n\) with minimum degree at least two, \( \gamma_{tdR}({G})\le\left\lfloor \frac{4n}{3}\right\rfloor \).