  We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in
C^\theta(\Omega)$, $\theta\in(0,1)$, of an inviscid incompressible fluid in a
bounded and simply connected domain $\Omega\subset \mathbb R^d$ with $C^{2+}$
boundary satisfies $p\in C^{\theta}(\Omega)$ for $\theta \leq \frac12$ and
$p\in C^{1,2\theta-1}(\Omega)$ for $\theta>\frac12$. Moreover, when $\partial
\Omega\in C^{3+}$, we prove that an almost double H\"older regularity $p\in
C^{2\theta-}(\Omega)$ holds even for $\theta<\frac12$. This extends and
improves the recent result of Bardos and Titi obtained in the planar case to
every dimension $d\ge2$ and it also doubles the pressure regularity.
Differently from Bardos and Titi, we do not introduce a new boundary condition
for the pressure, but instead work with the natural one. In the boundary-free
case of the $d$-dimensional torus, we show that the double regularity of the
pressure can be actually achieved under the weaker assumption that the
divergence of the velocity is sufficiently regular, thus not necessarily zero.
