Backbonding in phosphorous pentoxide
Coordinate bonds are covalent bonds where both bond electrons stem from only one of the two bond partners. They are formed when a Lewis base donates two electrons into accepting orbitals of the Lewis acid. That's not what you're looking at in this case, where both $\ce{P}$ and $\ce{O}$ contribute one electron to the common $\sigma$ bond. The interesting part is about everything that is to be described beyond the bond order of $1$. For example, everything that contributes to PO bonding beyond that single $\sigma$ bond.
I will further regard to phosphane oxides $\ce{P(=O)R3}$, where the question was originally asking about $\ce{P=O}$ bonds in $\ce{P4O10}$. The problem might be reduced to PO bonding in phosphoric acid, as $\ce{P4O10}$ is nothing more than its anhydride. The general observations regarding the $\ce{PO}$ bonding situation are unaffected. Kutzelnigg noticed in $1977$ already:

[The PO] bond is somewhere between single and triple and [...] only by chance it may happen to be a double bond.

What we know about the '$\ce{P=O}$ bond':

Its length is remarkably shorter than the P-O bond ($\pu{1.43 Å}$ compared to $\pu{1.60 Å}$ for $\ce{P4O10}$ according to Wikipedia).

It is highly polar.

Its bond strength of about $\pu{540 kJ/mol}$ is higher than for a corresponding single bond.


A critical review regarding the bonding situation in phosphane oxides was given by Gilheany. If it comes to Lewis valence formula, one might find three reasonable structures:

None of those is suitable to represent all theoretical and experimental findings regarding the characteristics of the $\ce{PO}$ bond!
Some scientists prefer to use the zwitterionic representation A, where the predominant $\sigma$ character of the $\ce{PO}$ bond is highlighted, while its dipolar character is also taken into account. If one chooses to write a Lewis structure as shown in B, one implies (besides one  $\sigma$ bond) the contribution of a $\pi$ bond between oxygen p orbitals and phosphorus d orbitals. However, the $3$d orbitals of phosphorus are no suitable $\pi$ accepting orbitals for $2$p orbitals of oxygen, as they are quite high in energy. Besides compatible symmetry, atomic orbitals (AOs) of the atoms in question require to be somewhat similar in energy to allow for a combination to form molecular orbitals (MOs).
If one allows $\pi$ back bonding from oxygen into $\sigma ^*$($\ce{P-R}$) (anti-bonding orbitals between P and its substituents R), one might come up with Lewis valence formula C. However, it should be noted that the $\pi$ bond order for $\ce{Me3PO}$ is only $0.7$ - and not $2$, what is implied by structure C!