  We prove that, if $\mathcal{GP}$ is the class of all Gorenstein projective
modules over a ring $R$, then $\mathfrak{GP}=(\mathcal{GP},\mathcal{GP}^\perp)$
is a cotorsion pair. Moreover, $\mathfrak{GP}$ is complete when all projective
modules are $\lambda$-pure-injective for some infinite regular cardinal
$\lambda$ (in particular, if $R$ is right $\Sigma$-pure-injective).
  We obtain these results, on the one hand, studying the class of totally
acyclic complexes over $R$. We prove that, when $R$ is $\Sigma$-pure-injective,
this class is deconstructible and forms a coreflective subcategory of the
homotopy category of the projective modules. On the other hand, we use results
about $\lambda$-pure-injective modules for infinite regular cardinals
$\lambda$.
  Finally, under different set-theoretical hypotheses, we show that for an
arbitrary ring $R$, the following hold: (1) There exists an infinite regular
cardinal number $\lambda$ such that every projective module is
$\lambda$-pure-injective (and $\mathfrak{GP}$ is complete). (2) $R$ is right
pure-semisimple if and only if there exists a regular uncountable $\lambda$
such that $\mathrm{Mod}$-$R$ has enough $\lambda$-pure-injective objects.
